Research ArticleHigh Accuracy 2D-DOA Estimation for Conformal ArrayUsing PARAFAC
Liangtian Wan1 Weijian Si1 Lutao Liu1 Zuoxi Tian2 and Naixing Feng3
1 Department of Information and Communication Engineering Harbin Engineering University Harbin 150001 China2 Science and Technology on Underwater Test and Control Laboratory Dalian 116013 China3 Institute of Electromagnetics and Acoustics Xiamen University Xiamen 361005 China
Correspondence should be addressed to Weijian Si swj0418263net
Received 26 October 2013 Revised 14 December 2013 Accepted 16 December 2013 Published 16 January 2014
Academic Editor Hon Tat Hui
Copyright copy 2014 Liangtian Wan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Due to the polarization diversity (PD) of element patterns caused by the varying curvature of the conformal carrier the conventionaldirection-of-arrival (DOA) estimation algorithms could not be applied to the conformal array In order to describe the PD ofconformal array the polarization parameter is considered in the snapshot datamodelThe paramount difficulty forDOA estimationis the coupling between the angle information and polarization parameter Based on the characteristic of the cylindrical conformalarray the decoupling between the polarization parameter and DOA can be realized with a specially designed array structure2D-DOA estimation of the cylindrical conformal array is accomplished via parallel factor analysis (PARAFAC) theory To avoidparameter pairing problem the algorithm forms a PARAFAC model of the covariance matrices in the covariance domain Theproposed algorithm can also be generalized to other conformal array structures and nonuniform noise scenario Cramer-Raobound (CRB) is derived and simulation results with the cylindrical conformal array are presented to verify the performance ofthe proposed algorithm
1 Introduction
Conformal arraysmounted on curved surfaces are commonlyapplied in various areas such as radar sonar and wirelesscommunication [1] The conformal antennas could fulfillspecific aerodynamics space-saving elimination of random-induced bore-sight error potential increase in available aper-ture and so on [2 3] Most researches focus on the designof antenna configuration [4ndash6] the transform between thelocal and global coordinate [7ndash10] and the pattern synthesisof conformal array [11ndash15]
The conventional direction-of-arrival (DOA) estimationalgorithms for example the multiple signal classification(MUSIC) [16] and the estimation of signal parameters viarotational invariance techniques (ESPRIT) [17] are notapplicable due to the varying curvature of the conformalarray The ldquoshadow effectrdquo of the conformal array is causedby the metallic shelter leading to the condition that notall elements have the ability to receive the signal As aresult of the incomplete steering vector common DOA
estimation algorithms cannot be used for conformal arrayRecently DOA estimation algorithms for conformal arrayhave been proposed for high resolution [18ndash22] With thehelp of fourth-order cumulant and ESPRIT algorithm a blindDOA estimation algorithm is proposed in [18] Based onthe mathematical technique of geometric algebra 2D-DOAand polarization parameter estimations are completed byexploiting iterative ESPRIT algorithm [19] The state spaceand propagator method are used for joint frequency and 2D-DOA estimations for the cylindrical conformal array [20]However the main drawback of the algorithms in [18ndash20] isthe need of parameter pairing
Parallel factor (PARAFAC) analysis attracted the atten-tion of researchers when it was originally introduced in arraysignal processing in 2000 [21 22] It has been widely usedfor low-rank decomposition of three and higher way arrayIn this paper a high accuracy 2D-DOA estimation algorithmfor the conformal array is proposed using PARAFAC theoryThe proposed algorithmdoes not need parameter pairing andperforms well when the signal-to-noise ratio (SNR) is low
Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 394707 14 pageshttpdxdoiorg1011552014394707
2 International Journal of Antennas and Propagation
or in situations where high estimation accuracy is neededSimulation results demonstrate the efficiency and accuracy ofthe proposed algorithm
The rest of this paper is structured as follows Section 2introduces the snapshot data model Section 3 elaborates thedesign of the conformal array structure Section 4 containsthe core contributions of this paper in which the PARAFACmodel is used for high accuracy 2D-DOA estimation Sec-tion 5 extends the proposed algorithm to more generalcases Section 6 analyses the computational complexity of thealgorithm Section 7 derives the Cramer-Rao bound (CRB)Section 8 discusses the array design Section 9 presents thesimulation result Conclusions are drawn in Section 10
2 The Snapshot Data Model
In order to achieve accurate parameter estimation the snap-shot data model of the conformal array must be preciselyestablished The pattern of the elements is different due tothe varying curvature of carriers [8] The mathematic modelof the conformal array constructed in [8 18] is adopted inthis paper The far-field narrowband signal impinging on theconformal array with elevation 120579 and azimuth 120593 is shown inFigure 1(a) The steering vector takes the following form
a (120579 120593) = [ℎ1119890minus1198952120587((p
1∙u)120582)
ℎ2119890minus1198952120587((p
2∙u)120582)
ℎ2119898+2
119890minus1198952120587((p
2119898+2∙u)120582)
]119879
(1)
ℎ119894= (119892
2
119894120579+ 119892
2
119894120593)12
(1198962
120579+ 119896
2
120593)12
cos (120579119894119892119896)
=1003816100381610038161003816119892119894100381610038161003816100381610038161003816100381610038161199011198971003816100381610038161003816 cos (120579119894119892119896)
= g119894∙ p
119897= 119892
119894120579119896120579+ 119892
119894120593119896120593
(2)
where P119894represents the position vector of the 119894th element u
is the propagation vector and 120582 is wavelength of the incidentsignal As shown in Figure 1(b) u
120579and u
120593are orthogonal unit
vectors and 119896120579and 119896
120593are the polarization components of the
incident signal ℎ119894denotes the 119894th elementrsquos response of the
incident signal g119894is the pattern of the 119894th element and p
119897is
the electric field direction of the incident signal 120579119894119892119896
is theangle between vector g
119894and p
119897 (∙)119879 denotes the transpose of
matrix (∙)Assuming that 119903 incident signals impinge on the designed
conformal array the snapshot data model can be representedas
X (119899) = G ∙ AS (119899) + N (119899)
= (G120579∙ A
120579K120579+ G
120593∙ A
120593K120593) S (119899) + N (119899)
= BS (119899) + N (119899)
G120579= [g
120579(1205791 120593
1) g
120579(1205792 120593
2) g
120579(120579119903 120593
119903)]
G120593= [g
120593(1205791 120593
1) g
120593(1205792 120593
2) g
120593(120579119903 120593
119903)]
A120579= [a
120579(1205791 120593
1) a
120579(1205792 120593
2) a
120579(120579119903 120593
119903)]
A120593= [a
120593(1205791 120593
1) a
120593(1205792 120593
2) a
120593(120579119903 120593
119903)]
K120579= diag (119896
1120579 119896
2120579 119896
119903120579)
K120593= diag (119896
1120593 119896
2120593 119896
119903120593)
S (119899) = [1199041(119899) 119904
2(119899) 119904
119903(119899)]
119879
W (119899) = [1199081(119899) 119908
2(119899) 119908
119903(119899)]
119879
(3)
where G denotes the pattern matrix and A denotes themanifold matrix K
120579and K
120593are the diagonal matrices whose
diagonal entries are 1198961120579 119896
2120579 119896
119903120579and 119896
1120593 119896
2120593 119896
119903120593
respectively 119896119894120579and 119896
119894120593are the components of 119894th signalrsquos
polarization parameters along the orthogonal unit vector u120579
and u120593 respectively S(119899) denotes the signal vector W(119899) is
additive Gaussian white noise with the covariance matrix
119864 W (119899)W(119899)119867 = Q = 120590
2I (4)
(∙)119867 denotes conjugate transpose of matrix (∙) Collecting119873
snapshots
X = BS +W (5)
where S is the 119903 times119873 signal matrix andW is the (2119898 + 2) times119873
noise matrixThe definition of the pattern is in the local coordinate
meaning that transformation must be performed from theglobal coordinate to the local coordinate [8] In addition thepolarization parameters couple with the signal parametersthus the decoupling is definitely necessary for the 2D-DOAestimation of the conformal array
3 The Structure of the Array
Since the signalrsquos amplitude of the sensors received willdecrease because of the ldquoshadow effectrdquo the whole conformalarray is divided into three subarrays and each subarray covers120
∘ For the single curvature and symmetry of the cylindereach subarray possesses and remains the same structure aswell as the parameter estimationmechanismThus this paperonly considers one subarray
As shown in Figure 2 the distance between two sensorswhich are in the same intersecting surface is 1205824 and thedistance between two adjacent intersecting surfaces is 5120582
As shown in Figure 2 1 sim 119898 constitute array 1 2 sim 119898 + 1
constitute array 2 119898 + 2 sim 2119898 + 1 constitute array 3 and119898 + 3 sim 2119898 + 2 constitute array 4 (1 sim 2119898 + 2 are sensorsof the array) The distance vector between array 1 and array2 is ΔP
1and 119889
1= |ΔP
1| = 1205824 The distance vector between
array 1 and array 3 is ΔP2and 119889
2= |ΔP
2| = 1205824 as shown
in Figure 3 The sensorsrsquo patterns of the same generatrix are
International Journal of Antennas and Propagation 3
Z
O
X
Y
120579
120593
minusu
(a)
g i
pl
O u120579k120579
gi120593
k120593
gi120579
120579igk
u120593
(b)
Figure 1 (a) The direction vector u impinges on the array (b) The 119894th sensors response
Z
Y
X
m + 1
m
O
2
1m + 2
m + 3
2m + 1
2m + 2
Figure 2 The structure of cylindrical conformal array
identical The sensors 1 sim 119898 + 1 possess the same patterng1 and the sensors119898 + 2 sim 2119898 + 2 possess the same pattern
g2The decoupling between polarization parameter and angle
information can be realized by taking full advantage of thecharacteristic of the array structure
Assume that X1 X
2 X
3 and X
4represent the data that
array 1 array 2 array 3 and array 4 receive respectively Theorigin of the coordinate is the reference point So the receiveddata can be expressed as
X1= BS + N
1
X2= BΨ
1S + N
2
X3= BΨ
2S + N
3
X4= BΨ
1Ψ2S + N
4
(6)
The covariance matrices among the received data are
R1= 119864 X
1X119867
1 = BR
119904B119867 +Q
1
R2= 119864 X
2X119867
1 = BΨ
1R119904B119867 +Q
2
R3= 119864 X
3X119867
1 = BΨ
2R119904B119867 +Q
3
R4= 119864 X
4X119867
1 = BΨ
1Ψ2R119904B119867 +Q
4
(7)
where R119904= diag1199042
1 119904
2
119903 represents the signal covariance
matrix and the noise covariance matrices areQ1ndashQ
4 respec-
tivelyConsider
Ψ1= diag [exp (minus119895120596
11) exp (minus119895120596
12) exp (minus119895120596
1119903)]
(8)
Ψ2= diag [
ℎ3(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
21)
ℎ3(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
22)
ℎ3(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
2119903)]
(9)
1205961119894= (
2120587
120582)ΔP
1∙ u
119894
= (2120587119889
1
120582)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(10)
1205962119894= (
2120587
120582)ΔP
2∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(11)
where 120579ΔP119894
and 120593ΔP119894
represent the elevation and azimuthof the distance vector in the global coordinate The array
4 International Journal of Antennas and Propagation
OY
Z
Δ
P2
P1
P1
(a)
Y
Xm+1 2m+1
O
ΔP
P
2
P
(b)
Figure 3 (a) The distance vector ΔP1 (b) The distance vector ΔP
2
structure is shown in Figures 2 and 3 in which the distancevectorΔP
1is parallel to the119885-axisThe elevation and azimuth
of ΔP1in the global coordinate are
120579ΔP1
= 0 120593ΔP1
=120587
2 (12)
ΔP2is parallel to the 119883-axis The elevation and azimuth of
ΔP2in the global coordinate are
120579ΔP2
=120587
2 120593
ΔP2
= 0 (13)
In real applications the ideal covariance matrix has to bereplaced by the sample covariance matrix which is obtainedby a finite number of snapshots R
119896(119896 = 1 4) that is
R119896=1
119873
119873
sum
119896=1
X (119899)X(119899)119867 =1
119873XXH
(14)
4 The DOA Estimation Based on PARAFAC
Parallel factor (PARAFAC) analysis is a method of multipledata analysis which has been first introduced in psycho-metrics It has been used in many fields such as statisticsarithmetic complexity and chemometrics In this sectionfirst the PARAFAC theory is introduced briefly secondbased on the PARAFAC model the multiple way array isconstructed and the identifiability of inherently unresolvablesource permutation and scaling ambiguities are analyzedfinally the trilinear alternating least squares (TALS) regres-sion algorithm is used to fit the PARAFAC model thusthe unknown parameters in the PARAFAC model can beestimated
41 Parallel Factor Analysis In this subsection the basicidea of PARAFAC theory is introduced First the trilineardecomposition of the element of the three-way array is elabo-rated Second based on the symmetrical characteristic of the
trilinear decomposition the three-way array is decomposedinto three dimensions
Considering a 119862 times 119863 times 119864 three-way array X (the entry ofthe array denotes 119909
119888119889119890) the trilinear decomposition of 119909
119888 119889119890
can be expressed as [21]
119909119888 119889 119890
=
119872
sum
119901=1
119904119888119901119905119889119901119906119890119901 (15)
where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 Thethree-way array X is represented as the sum of 119872 rank-1 factor Here the rank of X is defined as the minimumnumber of rank-1 three-way components which are neededto decompose X The vectors sp isin C119862times1 tp isin C119863times1 andup isin C119864times1 are called load vector score vector and factorprofiles respectively
Assume 119872 = 3 an example is given as follows andthus we can understand the idea of trilinear decompositionintuitivelyThe decomposition of three-way array X is shownin Figure 4 it can be seen that three dimensions can bedecomposed
The typical element of the 119862 times 119875 matrix S is defined asS(119888 119901) = 119904
119888119901 the typical element of the 119863 times 119875 matrix T is
defined as T(119888 119901) = 119905119888119901 the typical element of the 119864 times 119875
matrix U is defined as U(119888 119901) = 119906119888119901 In addition a 119863 times 119864
matrix X119888 a 119862 times 119864 matrix X
119889 and a 119862 times 119863 matrix X
119890are
defined The typical element of each matrix is X119888(119889 119890) =
X119889(119888 119890) = X
119890(119888 119889) = 119909
119888119889119890 The three-way matrix X can
be ldquoslicedrdquo along three different dimensions as shown inFigure 4
Consider
X119888= TΛ
119888(S)U119879
119888 = 1 119862
X119889= UΛ
119889(T) S119879 119889 = 1 119863
X119890= SΛ
119890(U)T119879 119890 = 1 119864
(16)
International Journal of Antennas and Propagation 5
= ++
1u 2u
3
3u
t2t1t
1s 2s 3sX
Figure 4 The decomposition of three-way array
where Λ119888(S) is the diagonal matrix constructed by the 119888th
row of the matrix S According to the definition of Khatri-Rao product X(119862119863times119864) can be written in another form as
X(119862119863times119864)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
X119888=1
X119888=2
X119888=119862
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
TΛ1(S)
TΛ2(S)
TΛ119862(S)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
U119879
= (S ⊙ T)UT (17)
where ldquo⊙rdquo is used to denote the Khatri-Rao (KR) product| ∙ | stands for the three-way array which is constructed bystacking thematrices into a columnThe definition of Khatri-Rao (KR) product can be found in the appendix The three-way array X can also be expressed in two other forms
X(119863119864times119862)= (T ⊙ U) ST (18)
X(119864119862times119863)= (U ⊙ S)TT
(19)
The reason of decomposition along three different dimen-sions is that the PARAFAC model can be solved by usingTALS algorithm The matrices T U and S can be updatedalternately
42 The Identifiability of the Model In this subsection thePARAFAC model is constructed by the received data ofthe conformal array The decomposition of the PARAFACmodel must be unique if not so the PARAFAC modelcannot be solved Thus some requirements of the matriceswhich construct the model have to be satisfied Theorem 2guarantees the uniqueness decomposition of themodel If thePARAFAC model is constructed by the received data of theconformal array then Theorem 2 can be used to judge theuniqueness decomposition of the model
On the basis of the PARAFAC theory [23] the 119898 times
119898 times 4 three-way array of the cylindrical conformal array isconstructed by (7)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q (20)
where R119894(119894 = 1 4) are the sample covariance matrices
respectively and Q represents the noise in real observation
Let C = B119867 based on the definition of Khatri-Rao product(20) can be transformed as
R = (D ⊙ B)C +Q (21)
R119883= (C119879 ⊙D)B119879 +Q
119883 (22)
R119884= (B ⊙ C119879)D119879
+Q119884 (23)
D =[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
]]]
]
(24)
where Λminus1(R119904) is the row vector constructed by the diagonal
entries of the diagonal matrix R119904 [∙] stands for the matrix
Definition 1 (see [21]) Considering a givenmatrixA isin C119862times119875if and only if the matrix A contains at least 119903 but not 119903 + 1linearly independent columns then the rank of matrix A is119903A = Rank(A) = 119903 If any 119896 columns of matrix A are linearlyindependent then the Kruskal rank (119896-rank) is 119896A = 119896 [24]Generally speaking 119896A le 119903A
In order to ensure the uniqueness of the PARAFACmodelin (21)ndash(23) the sufficient condition must be met
a ( 119894) = a ( 119895) (25)
For any two different incident signals 119904119894(119896) and 119904
119895(119896)
ℎ119900119890119890minus1198952120587((p
119900119890∙u119894)120582)
= ℎ119900119890119890minus1198952120587((p
119900119890∙u119895)120582) (26)
ensures the uniqueness of the PARAFAC model where
119901119900119890= sin (120579
119900119890) cos (120593
119900119890) x
+ sin (120579119900119890) sin (120593
119900119890) y + cos (120579
119900119890) z
(27)
u119894= sin (120579
119894) cos (120593
119894) x
+ sin (120579119894) sin (120593
119894) y + cos (120579
119894) z
(28)
120579119900119890
and 120593119900119890
are the elevation and azimuth of the sensorrsquosposition in the global coordinate 120579
119894and 120593
119894are the elevation
and azimuth of the 119894th incident signalThus (26) is equivalentto
sin (120579119900119890) cos (120593
119900119890) 120588
1
+ sin (120579119900119890) sin (120593
119900119890) 120588
2+ cos (120579
119900119890) 120588
3= 0
1205881= sin (120579
119894) cos (120593
119894) minus sin (120579
119895) cos (120593
119895)
1205882= sin (120579
119894) sin (120593
119894) minus sin (120579
119895) sin (120593
119895)
1205883= cos (120579
119894) minus cos (120579
119895)
(29)
Theorem 2 The sufficient conditions of the PARAFAC modelin (21)ndash(23) require that at least one of 120588
1and 120588
2and 120588
3is not
zero
6 International Journal of Antennas and Propagation
Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])
43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows
On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem
minDBC
R minus (D ⊙ B)C2119865 (30)
The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as
C = argminCR minus (D ⊙ B)C2
119865 (31)
Similarly the matrices B andD can be expressed as
B119879 = argminB
10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817
2
119865
D119879= argmin
D
10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817
2
119865
(32)
In the iterative procedure givenmatricesB andD thematrixC can be represented as
C = (D ⊙ B)daggerR (33)
The expression of matrices B119879 andD119879 is
B119879 = (C119879 ⊙D)dagger
R119883
D119879= (B ⊙ C119879)
dagger
R119884
(34)
where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as
follows
(1) initialize B(0) isin C119872times119875D(0)isin C4times119875
(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and
C by (35)ndash(37) Update just one matrix at each timethen 119896 rarr 119896 + 1
(4) else 120588(119896+1)minus120588(119896)120588(119896) lt 120576 the iteration is terminated
44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows
ThematrixD can be acquired by the TALS algorithm 1205961119894
and 1205962119894can be calculated by matrixD
1205961119894= minus
1
2(angle lceil
D2119894
D1119894
rceil + angle lceilD4119894
D3119894
rceil) (35)
where D119895119894represents the 119895th row of D lceil∙rceil stands for the
absolute value operation Because ℎ1and ℎ
3are real numbers
D3119894D
1119894and D
4119894D
2119894are squared to solve the ambiguity
caused by the positive and negative values of ℎ1and ℎ
3
Consider
1205962119894= minus
1
2angle([
ℎ3(120579119894 120593
119894)
ℎ1(120579119894 120593
119894)exp (minus119895120596
2119894)]
2
)
= minus1
2angle (exp (minus1198952120596
2119894))
= minus1
2angle([
D3119894
D1119894
]
2
)
= minus1
2angle([
D4119894
D2119894
]
2
)
(36)
Then
1205962119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D3119894
D1119894
]
2
) + angle([D4119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(37)
Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth
120593119894of 119894th incident signal can be expressed as
120579119894= arccos(
1205821205961119894
21205871198892
) = arccos(2120596
1119894
120587)
120593119894= arccos(
1205821205962119894
21205871198892sin (120579
119894)) = arccos(
21205962119894
120587 sin (120579119894))
(38)
Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically
Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows
(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
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RotatingMachinery
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
2 International Journal of Antennas and Propagation
or in situations where high estimation accuracy is neededSimulation results demonstrate the efficiency and accuracy ofthe proposed algorithm
The rest of this paper is structured as follows Section 2introduces the snapshot data model Section 3 elaborates thedesign of the conformal array structure Section 4 containsthe core contributions of this paper in which the PARAFACmodel is used for high accuracy 2D-DOA estimation Sec-tion 5 extends the proposed algorithm to more generalcases Section 6 analyses the computational complexity of thealgorithm Section 7 derives the Cramer-Rao bound (CRB)Section 8 discusses the array design Section 9 presents thesimulation result Conclusions are drawn in Section 10
2 The Snapshot Data Model
In order to achieve accurate parameter estimation the snap-shot data model of the conformal array must be preciselyestablished The pattern of the elements is different due tothe varying curvature of carriers [8] The mathematic modelof the conformal array constructed in [8 18] is adopted inthis paper The far-field narrowband signal impinging on theconformal array with elevation 120579 and azimuth 120593 is shown inFigure 1(a) The steering vector takes the following form
a (120579 120593) = [ℎ1119890minus1198952120587((p
1∙u)120582)
ℎ2119890minus1198952120587((p
2∙u)120582)
ℎ2119898+2
119890minus1198952120587((p
2119898+2∙u)120582)
]119879
(1)
ℎ119894= (119892
2
119894120579+ 119892
2
119894120593)12
(1198962
120579+ 119896
2
120593)12
cos (120579119894119892119896)
=1003816100381610038161003816119892119894100381610038161003816100381610038161003816100381610038161199011198971003816100381610038161003816 cos (120579119894119892119896)
= g119894∙ p
119897= 119892
119894120579119896120579+ 119892
119894120593119896120593
(2)
where P119894represents the position vector of the 119894th element u
is the propagation vector and 120582 is wavelength of the incidentsignal As shown in Figure 1(b) u
120579and u
120593are orthogonal unit
vectors and 119896120579and 119896
120593are the polarization components of the
incident signal ℎ119894denotes the 119894th elementrsquos response of the
incident signal g119894is the pattern of the 119894th element and p
119897is
the electric field direction of the incident signal 120579119894119892119896
is theangle between vector g
119894and p
119897 (∙)119879 denotes the transpose of
matrix (∙)Assuming that 119903 incident signals impinge on the designed
conformal array the snapshot data model can be representedas
X (119899) = G ∙ AS (119899) + N (119899)
= (G120579∙ A
120579K120579+ G
120593∙ A
120593K120593) S (119899) + N (119899)
= BS (119899) + N (119899)
G120579= [g
120579(1205791 120593
1) g
120579(1205792 120593
2) g
120579(120579119903 120593
119903)]
G120593= [g
120593(1205791 120593
1) g
120593(1205792 120593
2) g
120593(120579119903 120593
119903)]
A120579= [a
120579(1205791 120593
1) a
120579(1205792 120593
2) a
120579(120579119903 120593
119903)]
A120593= [a
120593(1205791 120593
1) a
120593(1205792 120593
2) a
120593(120579119903 120593
119903)]
K120579= diag (119896
1120579 119896
2120579 119896
119903120579)
K120593= diag (119896
1120593 119896
2120593 119896
119903120593)
S (119899) = [1199041(119899) 119904
2(119899) 119904
119903(119899)]
119879
W (119899) = [1199081(119899) 119908
2(119899) 119908
119903(119899)]
119879
(3)
where G denotes the pattern matrix and A denotes themanifold matrix K
120579and K
120593are the diagonal matrices whose
diagonal entries are 1198961120579 119896
2120579 119896
119903120579and 119896
1120593 119896
2120593 119896
119903120593
respectively 119896119894120579and 119896
119894120593are the components of 119894th signalrsquos
polarization parameters along the orthogonal unit vector u120579
and u120593 respectively S(119899) denotes the signal vector W(119899) is
additive Gaussian white noise with the covariance matrix
119864 W (119899)W(119899)119867 = Q = 120590
2I (4)
(∙)119867 denotes conjugate transpose of matrix (∙) Collecting119873
snapshots
X = BS +W (5)
where S is the 119903 times119873 signal matrix andW is the (2119898 + 2) times119873
noise matrixThe definition of the pattern is in the local coordinate
meaning that transformation must be performed from theglobal coordinate to the local coordinate [8] In addition thepolarization parameters couple with the signal parametersthus the decoupling is definitely necessary for the 2D-DOAestimation of the conformal array
3 The Structure of the Array
Since the signalrsquos amplitude of the sensors received willdecrease because of the ldquoshadow effectrdquo the whole conformalarray is divided into three subarrays and each subarray covers120
∘ For the single curvature and symmetry of the cylindereach subarray possesses and remains the same structure aswell as the parameter estimationmechanismThus this paperonly considers one subarray
As shown in Figure 2 the distance between two sensorswhich are in the same intersecting surface is 1205824 and thedistance between two adjacent intersecting surfaces is 5120582
As shown in Figure 2 1 sim 119898 constitute array 1 2 sim 119898 + 1
constitute array 2 119898 + 2 sim 2119898 + 1 constitute array 3 and119898 + 3 sim 2119898 + 2 constitute array 4 (1 sim 2119898 + 2 are sensorsof the array) The distance vector between array 1 and array2 is ΔP
1and 119889
1= |ΔP
1| = 1205824 The distance vector between
array 1 and array 3 is ΔP2and 119889
2= |ΔP
2| = 1205824 as shown
in Figure 3 The sensorsrsquo patterns of the same generatrix are
International Journal of Antennas and Propagation 3
Z
O
X
Y
120579
120593
minusu
(a)
g i
pl
O u120579k120579
gi120593
k120593
gi120579
120579igk
u120593
(b)
Figure 1 (a) The direction vector u impinges on the array (b) The 119894th sensors response
Z
Y
X
m + 1
m
O
2
1m + 2
m + 3
2m + 1
2m + 2
Figure 2 The structure of cylindrical conformal array
identical The sensors 1 sim 119898 + 1 possess the same patterng1 and the sensors119898 + 2 sim 2119898 + 2 possess the same pattern
g2The decoupling between polarization parameter and angle
information can be realized by taking full advantage of thecharacteristic of the array structure
Assume that X1 X
2 X
3 and X
4represent the data that
array 1 array 2 array 3 and array 4 receive respectively Theorigin of the coordinate is the reference point So the receiveddata can be expressed as
X1= BS + N
1
X2= BΨ
1S + N
2
X3= BΨ
2S + N
3
X4= BΨ
1Ψ2S + N
4
(6)
The covariance matrices among the received data are
R1= 119864 X
1X119867
1 = BR
119904B119867 +Q
1
R2= 119864 X
2X119867
1 = BΨ
1R119904B119867 +Q
2
R3= 119864 X
3X119867
1 = BΨ
2R119904B119867 +Q
3
R4= 119864 X
4X119867
1 = BΨ
1Ψ2R119904B119867 +Q
4
(7)
where R119904= diag1199042
1 119904
2
119903 represents the signal covariance
matrix and the noise covariance matrices areQ1ndashQ
4 respec-
tivelyConsider
Ψ1= diag [exp (minus119895120596
11) exp (minus119895120596
12) exp (minus119895120596
1119903)]
(8)
Ψ2= diag [
ℎ3(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
21)
ℎ3(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
22)
ℎ3(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
2119903)]
(9)
1205961119894= (
2120587
120582)ΔP
1∙ u
119894
= (2120587119889
1
120582)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(10)
1205962119894= (
2120587
120582)ΔP
2∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(11)
where 120579ΔP119894
and 120593ΔP119894
represent the elevation and azimuthof the distance vector in the global coordinate The array
4 International Journal of Antennas and Propagation
OY
Z
Δ
P2
P1
P1
(a)
Y
Xm+1 2m+1
O
ΔP
P
2
P
(b)
Figure 3 (a) The distance vector ΔP1 (b) The distance vector ΔP
2
structure is shown in Figures 2 and 3 in which the distancevectorΔP
1is parallel to the119885-axisThe elevation and azimuth
of ΔP1in the global coordinate are
120579ΔP1
= 0 120593ΔP1
=120587
2 (12)
ΔP2is parallel to the 119883-axis The elevation and azimuth of
ΔP2in the global coordinate are
120579ΔP2
=120587
2 120593
ΔP2
= 0 (13)
In real applications the ideal covariance matrix has to bereplaced by the sample covariance matrix which is obtainedby a finite number of snapshots R
119896(119896 = 1 4) that is
R119896=1
119873
119873
sum
119896=1
X (119899)X(119899)119867 =1
119873XXH
(14)
4 The DOA Estimation Based on PARAFAC
Parallel factor (PARAFAC) analysis is a method of multipledata analysis which has been first introduced in psycho-metrics It has been used in many fields such as statisticsarithmetic complexity and chemometrics In this sectionfirst the PARAFAC theory is introduced briefly secondbased on the PARAFAC model the multiple way array isconstructed and the identifiability of inherently unresolvablesource permutation and scaling ambiguities are analyzedfinally the trilinear alternating least squares (TALS) regres-sion algorithm is used to fit the PARAFAC model thusthe unknown parameters in the PARAFAC model can beestimated
41 Parallel Factor Analysis In this subsection the basicidea of PARAFAC theory is introduced First the trilineardecomposition of the element of the three-way array is elabo-rated Second based on the symmetrical characteristic of the
trilinear decomposition the three-way array is decomposedinto three dimensions
Considering a 119862 times 119863 times 119864 three-way array X (the entry ofthe array denotes 119909
119888119889119890) the trilinear decomposition of 119909
119888 119889119890
can be expressed as [21]
119909119888 119889 119890
=
119872
sum
119901=1
119904119888119901119905119889119901119906119890119901 (15)
where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 Thethree-way array X is represented as the sum of 119872 rank-1 factor Here the rank of X is defined as the minimumnumber of rank-1 three-way components which are neededto decompose X The vectors sp isin C119862times1 tp isin C119863times1 andup isin C119864times1 are called load vector score vector and factorprofiles respectively
Assume 119872 = 3 an example is given as follows andthus we can understand the idea of trilinear decompositionintuitivelyThe decomposition of three-way array X is shownin Figure 4 it can be seen that three dimensions can bedecomposed
The typical element of the 119862 times 119875 matrix S is defined asS(119888 119901) = 119904
119888119901 the typical element of the 119863 times 119875 matrix T is
defined as T(119888 119901) = 119905119888119901 the typical element of the 119864 times 119875
matrix U is defined as U(119888 119901) = 119906119888119901 In addition a 119863 times 119864
matrix X119888 a 119862 times 119864 matrix X
119889 and a 119862 times 119863 matrix X
119890are
defined The typical element of each matrix is X119888(119889 119890) =
X119889(119888 119890) = X
119890(119888 119889) = 119909
119888119889119890 The three-way matrix X can
be ldquoslicedrdquo along three different dimensions as shown inFigure 4
Consider
X119888= TΛ
119888(S)U119879
119888 = 1 119862
X119889= UΛ
119889(T) S119879 119889 = 1 119863
X119890= SΛ
119890(U)T119879 119890 = 1 119864
(16)
International Journal of Antennas and Propagation 5
= ++
1u 2u
3
3u
t2t1t
1s 2s 3sX
Figure 4 The decomposition of three-way array
where Λ119888(S) is the diagonal matrix constructed by the 119888th
row of the matrix S According to the definition of Khatri-Rao product X(119862119863times119864) can be written in another form as
X(119862119863times119864)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
X119888=1
X119888=2
X119888=119862
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
TΛ1(S)
TΛ2(S)
TΛ119862(S)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
U119879
= (S ⊙ T)UT (17)
where ldquo⊙rdquo is used to denote the Khatri-Rao (KR) product| ∙ | stands for the three-way array which is constructed bystacking thematrices into a columnThe definition of Khatri-Rao (KR) product can be found in the appendix The three-way array X can also be expressed in two other forms
X(119863119864times119862)= (T ⊙ U) ST (18)
X(119864119862times119863)= (U ⊙ S)TT
(19)
The reason of decomposition along three different dimen-sions is that the PARAFAC model can be solved by usingTALS algorithm The matrices T U and S can be updatedalternately
42 The Identifiability of the Model In this subsection thePARAFAC model is constructed by the received data ofthe conformal array The decomposition of the PARAFACmodel must be unique if not so the PARAFAC modelcannot be solved Thus some requirements of the matriceswhich construct the model have to be satisfied Theorem 2guarantees the uniqueness decomposition of themodel If thePARAFAC model is constructed by the received data of theconformal array then Theorem 2 can be used to judge theuniqueness decomposition of the model
On the basis of the PARAFAC theory [23] the 119898 times
119898 times 4 three-way array of the cylindrical conformal array isconstructed by (7)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q (20)
where R119894(119894 = 1 4) are the sample covariance matrices
respectively and Q represents the noise in real observation
Let C = B119867 based on the definition of Khatri-Rao product(20) can be transformed as
R = (D ⊙ B)C +Q (21)
R119883= (C119879 ⊙D)B119879 +Q
119883 (22)
R119884= (B ⊙ C119879)D119879
+Q119884 (23)
D =[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
]]]
]
(24)
where Λminus1(R119904) is the row vector constructed by the diagonal
entries of the diagonal matrix R119904 [∙] stands for the matrix
Definition 1 (see [21]) Considering a givenmatrixA isin C119862times119875if and only if the matrix A contains at least 119903 but not 119903 + 1linearly independent columns then the rank of matrix A is119903A = Rank(A) = 119903 If any 119896 columns of matrix A are linearlyindependent then the Kruskal rank (119896-rank) is 119896A = 119896 [24]Generally speaking 119896A le 119903A
In order to ensure the uniqueness of the PARAFACmodelin (21)ndash(23) the sufficient condition must be met
a ( 119894) = a ( 119895) (25)
For any two different incident signals 119904119894(119896) and 119904
119895(119896)
ℎ119900119890119890minus1198952120587((p
119900119890∙u119894)120582)
= ℎ119900119890119890minus1198952120587((p
119900119890∙u119895)120582) (26)
ensures the uniqueness of the PARAFAC model where
119901119900119890= sin (120579
119900119890) cos (120593
119900119890) x
+ sin (120579119900119890) sin (120593
119900119890) y + cos (120579
119900119890) z
(27)
u119894= sin (120579
119894) cos (120593
119894) x
+ sin (120579119894) sin (120593
119894) y + cos (120579
119894) z
(28)
120579119900119890
and 120593119900119890
are the elevation and azimuth of the sensorrsquosposition in the global coordinate 120579
119894and 120593
119894are the elevation
and azimuth of the 119894th incident signalThus (26) is equivalentto
sin (120579119900119890) cos (120593
119900119890) 120588
1
+ sin (120579119900119890) sin (120593
119900119890) 120588
2+ cos (120579
119900119890) 120588
3= 0
1205881= sin (120579
119894) cos (120593
119894) minus sin (120579
119895) cos (120593
119895)
1205882= sin (120579
119894) sin (120593
119894) minus sin (120579
119895) sin (120593
119895)
1205883= cos (120579
119894) minus cos (120579
119895)
(29)
Theorem 2 The sufficient conditions of the PARAFAC modelin (21)ndash(23) require that at least one of 120588
1and 120588
2and 120588
3is not
zero
6 International Journal of Antennas and Propagation
Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])
43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows
On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem
minDBC
R minus (D ⊙ B)C2119865 (30)
The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as
C = argminCR minus (D ⊙ B)C2
119865 (31)
Similarly the matrices B andD can be expressed as
B119879 = argminB
10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817
2
119865
D119879= argmin
D
10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817
2
119865
(32)
In the iterative procedure givenmatricesB andD thematrixC can be represented as
C = (D ⊙ B)daggerR (33)
The expression of matrices B119879 andD119879 is
B119879 = (C119879 ⊙D)dagger
R119883
D119879= (B ⊙ C119879)
dagger
R119884
(34)
where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as
follows
(1) initialize B(0) isin C119872times119875D(0)isin C4times119875
(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and
C by (35)ndash(37) Update just one matrix at each timethen 119896 rarr 119896 + 1
(4) else 120588(119896+1)minus120588(119896)120588(119896) lt 120576 the iteration is terminated
44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows
ThematrixD can be acquired by the TALS algorithm 1205961119894
and 1205962119894can be calculated by matrixD
1205961119894= minus
1
2(angle lceil
D2119894
D1119894
rceil + angle lceilD4119894
D3119894
rceil) (35)
where D119895119894represents the 119895th row of D lceil∙rceil stands for the
absolute value operation Because ℎ1and ℎ
3are real numbers
D3119894D
1119894and D
4119894D
2119894are squared to solve the ambiguity
caused by the positive and negative values of ℎ1and ℎ
3
Consider
1205962119894= minus
1
2angle([
ℎ3(120579119894 120593
119894)
ℎ1(120579119894 120593
119894)exp (minus119895120596
2119894)]
2
)
= minus1
2angle (exp (minus1198952120596
2119894))
= minus1
2angle([
D3119894
D1119894
]
2
)
= minus1
2angle([
D4119894
D2119894
]
2
)
(36)
Then
1205962119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D3119894
D1119894
]
2
) + angle([D4119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(37)
Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth
120593119894of 119894th incident signal can be expressed as
120579119894= arccos(
1205821205961119894
21205871198892
) = arccos(2120596
1119894
120587)
120593119894= arccos(
1205821205962119894
21205871198892sin (120579
119894)) = arccos(
21205962119894
120587 sin (120579119894))
(38)
Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically
Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows
(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 3
Z
O
X
Y
120579
120593
minusu
(a)
g i
pl
O u120579k120579
gi120593
k120593
gi120579
120579igk
u120593
(b)
Figure 1 (a) The direction vector u impinges on the array (b) The 119894th sensors response
Z
Y
X
m + 1
m
O
2
1m + 2
m + 3
2m + 1
2m + 2
Figure 2 The structure of cylindrical conformal array
identical The sensors 1 sim 119898 + 1 possess the same patterng1 and the sensors119898 + 2 sim 2119898 + 2 possess the same pattern
g2The decoupling between polarization parameter and angle
information can be realized by taking full advantage of thecharacteristic of the array structure
Assume that X1 X
2 X
3 and X
4represent the data that
array 1 array 2 array 3 and array 4 receive respectively Theorigin of the coordinate is the reference point So the receiveddata can be expressed as
X1= BS + N
1
X2= BΨ
1S + N
2
X3= BΨ
2S + N
3
X4= BΨ
1Ψ2S + N
4
(6)
The covariance matrices among the received data are
R1= 119864 X
1X119867
1 = BR
119904B119867 +Q
1
R2= 119864 X
2X119867
1 = BΨ
1R119904B119867 +Q
2
R3= 119864 X
3X119867
1 = BΨ
2R119904B119867 +Q
3
R4= 119864 X
4X119867
1 = BΨ
1Ψ2R119904B119867 +Q
4
(7)
where R119904= diag1199042
1 119904
2
119903 represents the signal covariance
matrix and the noise covariance matrices areQ1ndashQ
4 respec-
tivelyConsider
Ψ1= diag [exp (minus119895120596
11) exp (minus119895120596
12) exp (minus119895120596
1119903)]
(8)
Ψ2= diag [
ℎ3(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
21)
ℎ3(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
22)
ℎ3(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
2119903)]
(9)
1205961119894= (
2120587
120582)ΔP
1∙ u
119894
= (2120587119889
1
120582)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(10)
1205962119894= (
2120587
120582)ΔP
2∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(11)
where 120579ΔP119894
and 120593ΔP119894
represent the elevation and azimuthof the distance vector in the global coordinate The array
4 International Journal of Antennas and Propagation
OY
Z
Δ
P2
P1
P1
(a)
Y
Xm+1 2m+1
O
ΔP
P
2
P
(b)
Figure 3 (a) The distance vector ΔP1 (b) The distance vector ΔP
2
structure is shown in Figures 2 and 3 in which the distancevectorΔP
1is parallel to the119885-axisThe elevation and azimuth
of ΔP1in the global coordinate are
120579ΔP1
= 0 120593ΔP1
=120587
2 (12)
ΔP2is parallel to the 119883-axis The elevation and azimuth of
ΔP2in the global coordinate are
120579ΔP2
=120587
2 120593
ΔP2
= 0 (13)
In real applications the ideal covariance matrix has to bereplaced by the sample covariance matrix which is obtainedby a finite number of snapshots R
119896(119896 = 1 4) that is
R119896=1
119873
119873
sum
119896=1
X (119899)X(119899)119867 =1
119873XXH
(14)
4 The DOA Estimation Based on PARAFAC
Parallel factor (PARAFAC) analysis is a method of multipledata analysis which has been first introduced in psycho-metrics It has been used in many fields such as statisticsarithmetic complexity and chemometrics In this sectionfirst the PARAFAC theory is introduced briefly secondbased on the PARAFAC model the multiple way array isconstructed and the identifiability of inherently unresolvablesource permutation and scaling ambiguities are analyzedfinally the trilinear alternating least squares (TALS) regres-sion algorithm is used to fit the PARAFAC model thusthe unknown parameters in the PARAFAC model can beestimated
41 Parallel Factor Analysis In this subsection the basicidea of PARAFAC theory is introduced First the trilineardecomposition of the element of the three-way array is elabo-rated Second based on the symmetrical characteristic of the
trilinear decomposition the three-way array is decomposedinto three dimensions
Considering a 119862 times 119863 times 119864 three-way array X (the entry ofthe array denotes 119909
119888119889119890) the trilinear decomposition of 119909
119888 119889119890
can be expressed as [21]
119909119888 119889 119890
=
119872
sum
119901=1
119904119888119901119905119889119901119906119890119901 (15)
where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 Thethree-way array X is represented as the sum of 119872 rank-1 factor Here the rank of X is defined as the minimumnumber of rank-1 three-way components which are neededto decompose X The vectors sp isin C119862times1 tp isin C119863times1 andup isin C119864times1 are called load vector score vector and factorprofiles respectively
Assume 119872 = 3 an example is given as follows andthus we can understand the idea of trilinear decompositionintuitivelyThe decomposition of three-way array X is shownin Figure 4 it can be seen that three dimensions can bedecomposed
The typical element of the 119862 times 119875 matrix S is defined asS(119888 119901) = 119904
119888119901 the typical element of the 119863 times 119875 matrix T is
defined as T(119888 119901) = 119905119888119901 the typical element of the 119864 times 119875
matrix U is defined as U(119888 119901) = 119906119888119901 In addition a 119863 times 119864
matrix X119888 a 119862 times 119864 matrix X
119889 and a 119862 times 119863 matrix X
119890are
defined The typical element of each matrix is X119888(119889 119890) =
X119889(119888 119890) = X
119890(119888 119889) = 119909
119888119889119890 The three-way matrix X can
be ldquoslicedrdquo along three different dimensions as shown inFigure 4
Consider
X119888= TΛ
119888(S)U119879
119888 = 1 119862
X119889= UΛ
119889(T) S119879 119889 = 1 119863
X119890= SΛ
119890(U)T119879 119890 = 1 119864
(16)
International Journal of Antennas and Propagation 5
= ++
1u 2u
3
3u
t2t1t
1s 2s 3sX
Figure 4 The decomposition of three-way array
where Λ119888(S) is the diagonal matrix constructed by the 119888th
row of the matrix S According to the definition of Khatri-Rao product X(119862119863times119864) can be written in another form as
X(119862119863times119864)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
X119888=1
X119888=2
X119888=119862
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
TΛ1(S)
TΛ2(S)
TΛ119862(S)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
U119879
= (S ⊙ T)UT (17)
where ldquo⊙rdquo is used to denote the Khatri-Rao (KR) product| ∙ | stands for the three-way array which is constructed bystacking thematrices into a columnThe definition of Khatri-Rao (KR) product can be found in the appendix The three-way array X can also be expressed in two other forms
X(119863119864times119862)= (T ⊙ U) ST (18)
X(119864119862times119863)= (U ⊙ S)TT
(19)
The reason of decomposition along three different dimen-sions is that the PARAFAC model can be solved by usingTALS algorithm The matrices T U and S can be updatedalternately
42 The Identifiability of the Model In this subsection thePARAFAC model is constructed by the received data ofthe conformal array The decomposition of the PARAFACmodel must be unique if not so the PARAFAC modelcannot be solved Thus some requirements of the matriceswhich construct the model have to be satisfied Theorem 2guarantees the uniqueness decomposition of themodel If thePARAFAC model is constructed by the received data of theconformal array then Theorem 2 can be used to judge theuniqueness decomposition of the model
On the basis of the PARAFAC theory [23] the 119898 times
119898 times 4 three-way array of the cylindrical conformal array isconstructed by (7)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q (20)
where R119894(119894 = 1 4) are the sample covariance matrices
respectively and Q represents the noise in real observation
Let C = B119867 based on the definition of Khatri-Rao product(20) can be transformed as
R = (D ⊙ B)C +Q (21)
R119883= (C119879 ⊙D)B119879 +Q
119883 (22)
R119884= (B ⊙ C119879)D119879
+Q119884 (23)
D =[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
]]]
]
(24)
where Λminus1(R119904) is the row vector constructed by the diagonal
entries of the diagonal matrix R119904 [∙] stands for the matrix
Definition 1 (see [21]) Considering a givenmatrixA isin C119862times119875if and only if the matrix A contains at least 119903 but not 119903 + 1linearly independent columns then the rank of matrix A is119903A = Rank(A) = 119903 If any 119896 columns of matrix A are linearlyindependent then the Kruskal rank (119896-rank) is 119896A = 119896 [24]Generally speaking 119896A le 119903A
In order to ensure the uniqueness of the PARAFACmodelin (21)ndash(23) the sufficient condition must be met
a ( 119894) = a ( 119895) (25)
For any two different incident signals 119904119894(119896) and 119904
119895(119896)
ℎ119900119890119890minus1198952120587((p
119900119890∙u119894)120582)
= ℎ119900119890119890minus1198952120587((p
119900119890∙u119895)120582) (26)
ensures the uniqueness of the PARAFAC model where
119901119900119890= sin (120579
119900119890) cos (120593
119900119890) x
+ sin (120579119900119890) sin (120593
119900119890) y + cos (120579
119900119890) z
(27)
u119894= sin (120579
119894) cos (120593
119894) x
+ sin (120579119894) sin (120593
119894) y + cos (120579
119894) z
(28)
120579119900119890
and 120593119900119890
are the elevation and azimuth of the sensorrsquosposition in the global coordinate 120579
119894and 120593
119894are the elevation
and azimuth of the 119894th incident signalThus (26) is equivalentto
sin (120579119900119890) cos (120593
119900119890) 120588
1
+ sin (120579119900119890) sin (120593
119900119890) 120588
2+ cos (120579
119900119890) 120588
3= 0
1205881= sin (120579
119894) cos (120593
119894) minus sin (120579
119895) cos (120593
119895)
1205882= sin (120579
119894) sin (120593
119894) minus sin (120579
119895) sin (120593
119895)
1205883= cos (120579
119894) minus cos (120579
119895)
(29)
Theorem 2 The sufficient conditions of the PARAFAC modelin (21)ndash(23) require that at least one of 120588
1and 120588
2and 120588
3is not
zero
6 International Journal of Antennas and Propagation
Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])
43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows
On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem
minDBC
R minus (D ⊙ B)C2119865 (30)
The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as
C = argminCR minus (D ⊙ B)C2
119865 (31)
Similarly the matrices B andD can be expressed as
B119879 = argminB
10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817
2
119865
D119879= argmin
D
10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817
2
119865
(32)
In the iterative procedure givenmatricesB andD thematrixC can be represented as
C = (D ⊙ B)daggerR (33)
The expression of matrices B119879 andD119879 is
B119879 = (C119879 ⊙D)dagger
R119883
D119879= (B ⊙ C119879)
dagger
R119884
(34)
where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as
follows
(1) initialize B(0) isin C119872times119875D(0)isin C4times119875
(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and
C by (35)ndash(37) Update just one matrix at each timethen 119896 rarr 119896 + 1
(4) else 120588(119896+1)minus120588(119896)120588(119896) lt 120576 the iteration is terminated
44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows
ThematrixD can be acquired by the TALS algorithm 1205961119894
and 1205962119894can be calculated by matrixD
1205961119894= minus
1
2(angle lceil
D2119894
D1119894
rceil + angle lceilD4119894
D3119894
rceil) (35)
where D119895119894represents the 119895th row of D lceil∙rceil stands for the
absolute value operation Because ℎ1and ℎ
3are real numbers
D3119894D
1119894and D
4119894D
2119894are squared to solve the ambiguity
caused by the positive and negative values of ℎ1and ℎ
3
Consider
1205962119894= minus
1
2angle([
ℎ3(120579119894 120593
119894)
ℎ1(120579119894 120593
119894)exp (minus119895120596
2119894)]
2
)
= minus1
2angle (exp (minus1198952120596
2119894))
= minus1
2angle([
D3119894
D1119894
]
2
)
= minus1
2angle([
D4119894
D2119894
]
2
)
(36)
Then
1205962119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D3119894
D1119894
]
2
) + angle([D4119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(37)
Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth
120593119894of 119894th incident signal can be expressed as
120579119894= arccos(
1205821205961119894
21205871198892
) = arccos(2120596
1119894
120587)
120593119894= arccos(
1205821205962119894
21205871198892sin (120579
119894)) = arccos(
21205962119894
120587 sin (120579119894))
(38)
Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically
Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows
(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 International Journal of Antennas and Propagation
OY
Z
Δ
P2
P1
P1
(a)
Y
Xm+1 2m+1
O
ΔP
P
2
P
(b)
Figure 3 (a) The distance vector ΔP1 (b) The distance vector ΔP
2
structure is shown in Figures 2 and 3 in which the distancevectorΔP
1is parallel to the119885-axisThe elevation and azimuth
of ΔP1in the global coordinate are
120579ΔP1
= 0 120593ΔP1
=120587
2 (12)
ΔP2is parallel to the 119883-axis The elevation and azimuth of
ΔP2in the global coordinate are
120579ΔP2
=120587
2 120593
ΔP2
= 0 (13)
In real applications the ideal covariance matrix has to bereplaced by the sample covariance matrix which is obtainedby a finite number of snapshots R
119896(119896 = 1 4) that is
R119896=1
119873
119873
sum
119896=1
X (119899)X(119899)119867 =1
119873XXH
(14)
4 The DOA Estimation Based on PARAFAC
Parallel factor (PARAFAC) analysis is a method of multipledata analysis which has been first introduced in psycho-metrics It has been used in many fields such as statisticsarithmetic complexity and chemometrics In this sectionfirst the PARAFAC theory is introduced briefly secondbased on the PARAFAC model the multiple way array isconstructed and the identifiability of inherently unresolvablesource permutation and scaling ambiguities are analyzedfinally the trilinear alternating least squares (TALS) regres-sion algorithm is used to fit the PARAFAC model thusthe unknown parameters in the PARAFAC model can beestimated
41 Parallel Factor Analysis In this subsection the basicidea of PARAFAC theory is introduced First the trilineardecomposition of the element of the three-way array is elabo-rated Second based on the symmetrical characteristic of the
trilinear decomposition the three-way array is decomposedinto three dimensions
Considering a 119862 times 119863 times 119864 three-way array X (the entry ofthe array denotes 119909
119888119889119890) the trilinear decomposition of 119909
119888 119889119890
can be expressed as [21]
119909119888 119889 119890
=
119872
sum
119901=1
119904119888119901119905119889119901119906119890119901 (15)
where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 Thethree-way array X is represented as the sum of 119872 rank-1 factor Here the rank of X is defined as the minimumnumber of rank-1 three-way components which are neededto decompose X The vectors sp isin C119862times1 tp isin C119863times1 andup isin C119864times1 are called load vector score vector and factorprofiles respectively
Assume 119872 = 3 an example is given as follows andthus we can understand the idea of trilinear decompositionintuitivelyThe decomposition of three-way array X is shownin Figure 4 it can be seen that three dimensions can bedecomposed
The typical element of the 119862 times 119875 matrix S is defined asS(119888 119901) = 119904
119888119901 the typical element of the 119863 times 119875 matrix T is
defined as T(119888 119901) = 119905119888119901 the typical element of the 119864 times 119875
matrix U is defined as U(119888 119901) = 119906119888119901 In addition a 119863 times 119864
matrix X119888 a 119862 times 119864 matrix X
119889 and a 119862 times 119863 matrix X
119890are
defined The typical element of each matrix is X119888(119889 119890) =
X119889(119888 119890) = X
119890(119888 119889) = 119909
119888119889119890 The three-way matrix X can
be ldquoslicedrdquo along three different dimensions as shown inFigure 4
Consider
X119888= TΛ
119888(S)U119879
119888 = 1 119862
X119889= UΛ
119889(T) S119879 119889 = 1 119863
X119890= SΛ
119890(U)T119879 119890 = 1 119864
(16)
International Journal of Antennas and Propagation 5
= ++
1u 2u
3
3u
t2t1t
1s 2s 3sX
Figure 4 The decomposition of three-way array
where Λ119888(S) is the diagonal matrix constructed by the 119888th
row of the matrix S According to the definition of Khatri-Rao product X(119862119863times119864) can be written in another form as
X(119862119863times119864)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
X119888=1
X119888=2
X119888=119862
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
TΛ1(S)
TΛ2(S)
TΛ119862(S)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
U119879
= (S ⊙ T)UT (17)
where ldquo⊙rdquo is used to denote the Khatri-Rao (KR) product| ∙ | stands for the three-way array which is constructed bystacking thematrices into a columnThe definition of Khatri-Rao (KR) product can be found in the appendix The three-way array X can also be expressed in two other forms
X(119863119864times119862)= (T ⊙ U) ST (18)
X(119864119862times119863)= (U ⊙ S)TT
(19)
The reason of decomposition along three different dimen-sions is that the PARAFAC model can be solved by usingTALS algorithm The matrices T U and S can be updatedalternately
42 The Identifiability of the Model In this subsection thePARAFAC model is constructed by the received data ofthe conformal array The decomposition of the PARAFACmodel must be unique if not so the PARAFAC modelcannot be solved Thus some requirements of the matriceswhich construct the model have to be satisfied Theorem 2guarantees the uniqueness decomposition of themodel If thePARAFAC model is constructed by the received data of theconformal array then Theorem 2 can be used to judge theuniqueness decomposition of the model
On the basis of the PARAFAC theory [23] the 119898 times
119898 times 4 three-way array of the cylindrical conformal array isconstructed by (7)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q (20)
where R119894(119894 = 1 4) are the sample covariance matrices
respectively and Q represents the noise in real observation
Let C = B119867 based on the definition of Khatri-Rao product(20) can be transformed as
R = (D ⊙ B)C +Q (21)
R119883= (C119879 ⊙D)B119879 +Q
119883 (22)
R119884= (B ⊙ C119879)D119879
+Q119884 (23)
D =[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
]]]
]
(24)
where Λminus1(R119904) is the row vector constructed by the diagonal
entries of the diagonal matrix R119904 [∙] stands for the matrix
Definition 1 (see [21]) Considering a givenmatrixA isin C119862times119875if and only if the matrix A contains at least 119903 but not 119903 + 1linearly independent columns then the rank of matrix A is119903A = Rank(A) = 119903 If any 119896 columns of matrix A are linearlyindependent then the Kruskal rank (119896-rank) is 119896A = 119896 [24]Generally speaking 119896A le 119903A
In order to ensure the uniqueness of the PARAFACmodelin (21)ndash(23) the sufficient condition must be met
a ( 119894) = a ( 119895) (25)
For any two different incident signals 119904119894(119896) and 119904
119895(119896)
ℎ119900119890119890minus1198952120587((p
119900119890∙u119894)120582)
= ℎ119900119890119890minus1198952120587((p
119900119890∙u119895)120582) (26)
ensures the uniqueness of the PARAFAC model where
119901119900119890= sin (120579
119900119890) cos (120593
119900119890) x
+ sin (120579119900119890) sin (120593
119900119890) y + cos (120579
119900119890) z
(27)
u119894= sin (120579
119894) cos (120593
119894) x
+ sin (120579119894) sin (120593
119894) y + cos (120579
119894) z
(28)
120579119900119890
and 120593119900119890
are the elevation and azimuth of the sensorrsquosposition in the global coordinate 120579
119894and 120593
119894are the elevation
and azimuth of the 119894th incident signalThus (26) is equivalentto
sin (120579119900119890) cos (120593
119900119890) 120588
1
+ sin (120579119900119890) sin (120593
119900119890) 120588
2+ cos (120579
119900119890) 120588
3= 0
1205881= sin (120579
119894) cos (120593
119894) minus sin (120579
119895) cos (120593
119895)
1205882= sin (120579
119894) sin (120593
119894) minus sin (120579
119895) sin (120593
119895)
1205883= cos (120579
119894) minus cos (120579
119895)
(29)
Theorem 2 The sufficient conditions of the PARAFAC modelin (21)ndash(23) require that at least one of 120588
1and 120588
2and 120588
3is not
zero
6 International Journal of Antennas and Propagation
Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])
43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows
On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem
minDBC
R minus (D ⊙ B)C2119865 (30)
The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as
C = argminCR minus (D ⊙ B)C2
119865 (31)
Similarly the matrices B andD can be expressed as
B119879 = argminB
10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817
2
119865
D119879= argmin
D
10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817
2
119865
(32)
In the iterative procedure givenmatricesB andD thematrixC can be represented as
C = (D ⊙ B)daggerR (33)
The expression of matrices B119879 andD119879 is
B119879 = (C119879 ⊙D)dagger
R119883
D119879= (B ⊙ C119879)
dagger
R119884
(34)
where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as
follows
(1) initialize B(0) isin C119872times119875D(0)isin C4times119875
(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and
C by (35)ndash(37) Update just one matrix at each timethen 119896 rarr 119896 + 1
(4) else 120588(119896+1)minus120588(119896)120588(119896) lt 120576 the iteration is terminated
44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows
ThematrixD can be acquired by the TALS algorithm 1205961119894
and 1205962119894can be calculated by matrixD
1205961119894= minus
1
2(angle lceil
D2119894
D1119894
rceil + angle lceilD4119894
D3119894
rceil) (35)
where D119895119894represents the 119895th row of D lceil∙rceil stands for the
absolute value operation Because ℎ1and ℎ
3are real numbers
D3119894D
1119894and D
4119894D
2119894are squared to solve the ambiguity
caused by the positive and negative values of ℎ1and ℎ
3
Consider
1205962119894= minus
1
2angle([
ℎ3(120579119894 120593
119894)
ℎ1(120579119894 120593
119894)exp (minus119895120596
2119894)]
2
)
= minus1
2angle (exp (minus1198952120596
2119894))
= minus1
2angle([
D3119894
D1119894
]
2
)
= minus1
2angle([
D4119894
D2119894
]
2
)
(36)
Then
1205962119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D3119894
D1119894
]
2
) + angle([D4119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(37)
Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth
120593119894of 119894th incident signal can be expressed as
120579119894= arccos(
1205821205961119894
21205871198892
) = arccos(2120596
1119894
120587)
120593119894= arccos(
1205821205962119894
21205871198892sin (120579
119894)) = arccos(
21205962119894
120587 sin (120579119894))
(38)
Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically
Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows
(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 5
= ++
1u 2u
3
3u
t2t1t
1s 2s 3sX
Figure 4 The decomposition of three-way array
where Λ119888(S) is the diagonal matrix constructed by the 119888th
row of the matrix S According to the definition of Khatri-Rao product X(119862119863times119864) can be written in another form as
X(119862119863times119864)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
X119888=1
X119888=2
X119888=119862
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
TΛ1(S)
TΛ2(S)
TΛ119862(S)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
U119879
= (S ⊙ T)UT (17)
where ldquo⊙rdquo is used to denote the Khatri-Rao (KR) product| ∙ | stands for the three-way array which is constructed bystacking thematrices into a columnThe definition of Khatri-Rao (KR) product can be found in the appendix The three-way array X can also be expressed in two other forms
X(119863119864times119862)= (T ⊙ U) ST (18)
X(119864119862times119863)= (U ⊙ S)TT
(19)
The reason of decomposition along three different dimen-sions is that the PARAFAC model can be solved by usingTALS algorithm The matrices T U and S can be updatedalternately
42 The Identifiability of the Model In this subsection thePARAFAC model is constructed by the received data ofthe conformal array The decomposition of the PARAFACmodel must be unique if not so the PARAFAC modelcannot be solved Thus some requirements of the matriceswhich construct the model have to be satisfied Theorem 2guarantees the uniqueness decomposition of themodel If thePARAFAC model is constructed by the received data of theconformal array then Theorem 2 can be used to judge theuniqueness decomposition of the model
On the basis of the PARAFAC theory [23] the 119898 times
119898 times 4 three-way array of the cylindrical conformal array isconstructed by (7)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q (20)
where R119894(119894 = 1 4) are the sample covariance matrices
respectively and Q represents the noise in real observation
Let C = B119867 based on the definition of Khatri-Rao product(20) can be transformed as
R = (D ⊙ B)C +Q (21)
R119883= (C119879 ⊙D)B119879 +Q
119883 (22)
R119884= (B ⊙ C119879)D119879
+Q119884 (23)
D =[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
]]]
]
(24)
where Λminus1(R119904) is the row vector constructed by the diagonal
entries of the diagonal matrix R119904 [∙] stands for the matrix
Definition 1 (see [21]) Considering a givenmatrixA isin C119862times119875if and only if the matrix A contains at least 119903 but not 119903 + 1linearly independent columns then the rank of matrix A is119903A = Rank(A) = 119903 If any 119896 columns of matrix A are linearlyindependent then the Kruskal rank (119896-rank) is 119896A = 119896 [24]Generally speaking 119896A le 119903A
In order to ensure the uniqueness of the PARAFACmodelin (21)ndash(23) the sufficient condition must be met
a ( 119894) = a ( 119895) (25)
For any two different incident signals 119904119894(119896) and 119904
119895(119896)
ℎ119900119890119890minus1198952120587((p
119900119890∙u119894)120582)
= ℎ119900119890119890minus1198952120587((p
119900119890∙u119895)120582) (26)
ensures the uniqueness of the PARAFAC model where
119901119900119890= sin (120579
119900119890) cos (120593
119900119890) x
+ sin (120579119900119890) sin (120593
119900119890) y + cos (120579
119900119890) z
(27)
u119894= sin (120579
119894) cos (120593
119894) x
+ sin (120579119894) sin (120593
119894) y + cos (120579
119894) z
(28)
120579119900119890
and 120593119900119890
are the elevation and azimuth of the sensorrsquosposition in the global coordinate 120579
119894and 120593
119894are the elevation
and azimuth of the 119894th incident signalThus (26) is equivalentto
sin (120579119900119890) cos (120593
119900119890) 120588
1
+ sin (120579119900119890) sin (120593
119900119890) 120588
2+ cos (120579
119900119890) 120588
3= 0
1205881= sin (120579
119894) cos (120593
119894) minus sin (120579
119895) cos (120593
119895)
1205882= sin (120579
119894) sin (120593
119894) minus sin (120579
119895) sin (120593
119895)
1205883= cos (120579
119894) minus cos (120579
119895)
(29)
Theorem 2 The sufficient conditions of the PARAFAC modelin (21)ndash(23) require that at least one of 120588
1and 120588
2and 120588
3is not
zero
6 International Journal of Antennas and Propagation
Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])
43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows
On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem
minDBC
R minus (D ⊙ B)C2119865 (30)
The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as
C = argminCR minus (D ⊙ B)C2
119865 (31)
Similarly the matrices B andD can be expressed as
B119879 = argminB
10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817
2
119865
D119879= argmin
D
10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817
2
119865
(32)
In the iterative procedure givenmatricesB andD thematrixC can be represented as
C = (D ⊙ B)daggerR (33)
The expression of matrices B119879 andD119879 is
B119879 = (C119879 ⊙D)dagger
R119883
D119879= (B ⊙ C119879)
dagger
R119884
(34)
where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as
follows
(1) initialize B(0) isin C119872times119875D(0)isin C4times119875
(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and
C by (35)ndash(37) Update just one matrix at each timethen 119896 rarr 119896 + 1
(4) else 120588(119896+1)minus120588(119896)120588(119896) lt 120576 the iteration is terminated
44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows
ThematrixD can be acquired by the TALS algorithm 1205961119894
and 1205962119894can be calculated by matrixD
1205961119894= minus
1
2(angle lceil
D2119894
D1119894
rceil + angle lceilD4119894
D3119894
rceil) (35)
where D119895119894represents the 119895th row of D lceil∙rceil stands for the
absolute value operation Because ℎ1and ℎ
3are real numbers
D3119894D
1119894and D
4119894D
2119894are squared to solve the ambiguity
caused by the positive and negative values of ℎ1and ℎ
3
Consider
1205962119894= minus
1
2angle([
ℎ3(120579119894 120593
119894)
ℎ1(120579119894 120593
119894)exp (minus119895120596
2119894)]
2
)
= minus1
2angle (exp (minus1198952120596
2119894))
= minus1
2angle([
D3119894
D1119894
]
2
)
= minus1
2angle([
D4119894
D2119894
]
2
)
(36)
Then
1205962119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D3119894
D1119894
]
2
) + angle([D4119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(37)
Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth
120593119894of 119894th incident signal can be expressed as
120579119894= arccos(
1205821205961119894
21205871198892
) = arccos(2120596
1119894
120587)
120593119894= arccos(
1205821205962119894
21205871198892sin (120579
119894)) = arccos(
21205962119894
120587 sin (120579119894))
(38)
Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically
Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows
(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 International Journal of Antennas and Propagation
Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])
43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows
On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem
minDBC
R minus (D ⊙ B)C2119865 (30)
The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as
C = argminCR minus (D ⊙ B)C2
119865 (31)
Similarly the matrices B andD can be expressed as
B119879 = argminB
10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817
2
119865
D119879= argmin
D
10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817
2
119865
(32)
In the iterative procedure givenmatricesB andD thematrixC can be represented as
C = (D ⊙ B)daggerR (33)
The expression of matrices B119879 andD119879 is
B119879 = (C119879 ⊙D)dagger
R119883
D119879= (B ⊙ C119879)
dagger
R119884
(34)
where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as
follows
(1) initialize B(0) isin C119872times119875D(0)isin C4times119875
(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and
C by (35)ndash(37) Update just one matrix at each timethen 119896 rarr 119896 + 1
(4) else 120588(119896+1)minus120588(119896)120588(119896) lt 120576 the iteration is terminated
44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows
ThematrixD can be acquired by the TALS algorithm 1205961119894
and 1205962119894can be calculated by matrixD
1205961119894= minus
1
2(angle lceil
D2119894
D1119894
rceil + angle lceilD4119894
D3119894
rceil) (35)
where D119895119894represents the 119895th row of D lceil∙rceil stands for the
absolute value operation Because ℎ1and ℎ
3are real numbers
D3119894D
1119894and D
4119894D
2119894are squared to solve the ambiguity
caused by the positive and negative values of ℎ1and ℎ
3
Consider
1205962119894= minus
1
2angle([
ℎ3(120579119894 120593
119894)
ℎ1(120579119894 120593
119894)exp (minus119895120596
2119894)]
2
)
= minus1
2angle (exp (minus1198952120596
2119894))
= minus1
2angle([
D3119894
D1119894
]
2
)
= minus1
2angle([
D4119894
D2119894
]
2
)
(36)
Then
1205962119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D3119894
D1119894
]
2
) + angle([D4119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(37)
Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth
120593119894of 119894th incident signal can be expressed as
120579119894= arccos(
1205821205961119894
21205871198892
) = arccos(2120596
1119894
120587)
120593119894= arccos(
1205821205962119894
21205871198892sin (120579
119894)) = arccos(
21205962119894
120587 sin (120579119894))
(38)
Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically
Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows
(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
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International Journal of
International Journal of Antennas and Propagation 7
Z
Y
X
m + 1
O
m
2m + 1
1
2m + 2
2m + 3
2m + 4
3m + 2
3m + 3
2m + 2
m + 3
Figure 5 The extendable cylindrical conformal array structure
(2) construct the PARAFAC model by (20)ndash(23)
(3) estimate the matrixD using the TALS algorithm
(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596
1119894and 120596
2119894
(5) acquire the elevation and azimuth estimation from(40) and (41)
5 The Extendable Array Structure
The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array
51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained
As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP
2between array 1 and
array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3
is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g
3 The distance vector between array 1 and array 5 is
ΔP3 and 119889
3= |ΔP
3| Under the design in Figure 5 the only
restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are
X5= BΨ
3S + N
5 (39)
X6= BΨ
1Ψ3S + N
6 (40)
where
Ψ3= diag [
ℎ5(1205791 120593
1)
ℎ1(1205791 120593
1)exp (minus119895120596
31)
ℎ5(1205792 120593
2)
ℎ1(1205792 120593
2)exp (minus119895120596
32)
ℎ5(120579119903 120593
119903)
ℎ1(120579119903 120593
119903)exp (minus119895120596
3119903)]
(41)
1205963119894= (
2120587
120582)ΔP
3∙ u
119894
= (2120587119889
2
120582)
times [sin (120579ΔP3
) cos (120593ΔP3
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP3
) cos (120579119894)]
(42)
The cross-covariance matrices among array 1 array 5 andarray 6 are
R5= 119864 X
5X119867
1 = BΨ
3R119904B119867 +Q
5
R6= 119864 X
6X119867
1 = BΨ
1Ψ3R119904B119867 +Q
6
(43)
where Q5and Q
6are noise covariance matrices Then (20)
is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
R5
R6
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
BR119904B119867
BΨ1R119904B119867
BΨ2R119904B119867
BΨ1Ψ2R119904B119867
BΨ3R119904B119867
BΨ1Ψ3R119904B119867
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q1 (44)
where Q1is the observed noise The form of Khatri-Rao
product is applied in (44)
R = (D ⊙ B)C +Q1 (45)
where
D =
[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
1Ψ2R119904)
Λminus1(Ψ
3R119904)
Λminus1(Ψ
1Ψ3R119904)
]]]]]]]
]
(46)
As long as Theorem 2 holds (45) is unique 1205963119894can be
obtained similarly with (37)
1205963119894= minus
1
4
100381610038161003816100381610038161003816100381610038161003816
angle([D5119894
D1119894
]
2
) + angle([D6119894
D2119894
]
2
)
100381610038161003816100381610038161003816100381610038161003816
(47)
after calculating matrixD by ALS algorithm
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 International Journal of Antennas and Propagation
Assume that Δ11990111
= sin(120579ΔP1
) cos(120593ΔP1
) Δ11990112
=
sin(120579ΔP1
) sin(120593ΔP1
) andΔ11990113= cos(120579
ΔP1
) similar toΔ1199012119894and
Δ1199013119894(119894 = 1 2 3) 120574
1119894= sin(120579
119894) cos(120593
119894) 120574
2119894= sin(120579
119894) sin(120593
119894)
and 1205743119894= cos(120579
119894) solving (10) (11) (42) (36) (37) and (24)
we can obtain
120582
2120587
[[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
= [
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
[
[
1205741119894
1205742119894
1205743119894
]
]
(48)
The solution of (48) is
[
[
1205741119894
1205742119894
1205743119894
]
]
=120582
2120587
[
[
Δ11990111
Δ11990112
Δ11990113
Δ11990121
Δ11990122
Δ11990123
Δ11990131
Δ11990132
Δ11990133
]
]
minus1 [[[[[
[
1205961119894
1198891
1205962119894
1198892
1205963119894
1198893
]]]]]
]
(49)
1205741 120574
2 and 120574
3can be acquired by solving (49) Two methods
can be used to obtain 120579119894120593
119894The firstmethod can be described
as
120579119894= arccos 120574
3119894
120593119894= arcsin(
1205741119894
cos (120579119894)) or 120593
119894= arcsin(
1205742119894
sin (120579119894))
(50)
The second one is
120593119894= arctan(
1205742119894
1205741119894
)
120579119894= arccos(
1205741119894
cos (120593119894)) or 120579
119894= arcsin(
1205742119894
sin (120593119894))
(51)
52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution
Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP
1 and
1198891= |ΔP
1| The distance vector between array 3 and array
4 is ΔP2 and 119889
2= |ΔP
2| 1 sim 119898 + 1 possess the same
pattern g1 and 1 sim 2119898 + 1 possess the same pattern g
2
The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure
Z
Y
XO
3m2m
2m + 1
2m + 2
2m + 3
3m + 1
m + 2
1
2
m
m
+ 3 3
m + 1
Figure 6 The extendable conical conformal array structure
The received data from array 1 to array 4 are
X1= B
1S + N
1 (52)
X2= B
1Ψ1S + N
2 (53)
X3= B
2S + N
3 (54)
X4= B
2Ψ2S + N
4 (55)
where N1ndashN
6are the noise matrices received by each array
The cross-covariance matrices among the arrays are
R1= 119864 X
3X119867
1 = B
2R119904B1198671+Q
1
R2= 119864 X
4X119867
1 = B
2Ψ2R119904B1198671+Q
2
R3= 119864 X
3X119867
2 = B
2Ψ119867
1R119904B1198671+Q
3
R4= 119864 X
4X119867
2 = B
2Ψ119867
1Ψ2R119904B1198671+Q
4
(56)
where B1is the manifold matrix of array 1 and array 2 and
B2is the manifold matrix of array 3 and array 4 Q
1ndashQ
4
are the noise covariance matrices received by the arrays Ψ1
and Ψ2possess the same forms as (8) and (9) According to
PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R ( 1)R ( 2)R ( 3)R ( 4)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
R1
R2
R3
R4
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
B2R119904B1198671
B2Ψ119867
1R119904B1198671
B2Ψ2R119904B1198671
B2Ψ119867
1Ψ2R119904B1198671
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
+Q2
(57)
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 9
and Q2is the observed noise Let C = B119867
1 and (57) can be
written in the following form of Khatri-Rao product
R = (D ⊙ B2)C +Q
2
D =
[[[[[[[[
[
Λminus1(R
119904)
Λminus1(Ψ
119867
1R119904)
Λminus1(Ψ
2R119904)
Λminus1(Ψ
119867
1Ψ2R119904)
]]]]]]]]
]
(58)
The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596
1119894and 120596
2119894can be represented as
1205961119894=1
2(angle[
[[
D2119894
D1119894
]]]
+ angle[[[
D4119894
D3119894
]]]
) (59)
1205962119894= minus
1
2(angle[
[[
D3119894
D1119894
]]]
+ angle[[[
D4119894
D2119894
]]]
) (60)
1205961119894= (
2120587
120582119894
)ΔP1∙ u
119894
= (2120587119889
1
120582119894
)
times [sin (120579ΔP1
) cos (120593ΔP1
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP1
) sin (120593ΔP1
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP1
) cos (120579119894)]
(61)
1205962119894= (
2120587
120582119894
)ΔP2∙ u
119894
= (2120587119889
2
120582119894
)
times [sin (120579ΔP2
) cos (120593ΔP2
) sin (120579119894) cos (120593
119894)
+ sin (120579ΔP2
) sin (120593ΔP2
) sin (120579119894) sin (120593
119894)
+ cos (120579ΔP2
) cos (120579119894)]
(62)
The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as
1198911(119909
1 119909
2) = 0
1198912(119909
1 119909
2) = 0
(63)
where 1199091= 120579 and 119909
2= 120593 The equation with the same
solution of (63) is expressed as
x = 119891minus1119894(119909
1 119909
2) 119894 = 1 2 (64)
The iterative form can be represented as
x119896+1 = 119865119894(119909
119896
1 119909
119896
2) 119894 = 1 2 (65)
The initial vector is selected as x(0) = [119909(0)1 119909
(0)
2]119879
Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal
Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed
6 The Computational Complexity Analysis
We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898
matrices requires about119874(241198983)The algorithm in this paper
uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898
2119903) For the simulation in Section 9 the proposed
algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903
3+4119898
2119903)] (119870 stands for the number of iterations) Since
the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity
7 CRB of Joint Polarization andDOA Estimation
Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R
119904is assumed to be known and the
noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as
k119879 = [1205791 120593
1 1205792 120593
2 120579
119903 120593
119903] (66)
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 International Journal of Antennas and Propagation
The CRB of joint polarization parameters and angle parame-ters is defined as
119864 [(k minus k) (k minus k)119879] ge CRB
CRB = Fminus1(67)
The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by
F = [F120579120579 F120579120593
F120593120579
F120593120593
] (68)
where F120579120579
is the block matrix of elevation estimator and F120593120593
is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as
F119894119895= 119873trace[Rminus1 120597R
120597V119894
Rminus1 120597R120597V
119895
]
= 2119873Re trace [D119894R119904B119867Rminus1BR
119904D119867
119895Rminus1]
+ trace [D119894R119904B119867Rminus1BR
119904D119895Rminus1]
D119894=120597B120597k
119894
(69)
where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the
partial derivative of matrix R
8 Discussion
The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution
(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP
1is perpendicular to ΔP
2 This is
one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP
1or ΔP
2is parallel
or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows
The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP
1 The
distance vector between array 3 and array 4 is ΔP2 As shown
in Figures 7 and 8 the distance vector ΔP1is perpendicular
to ΔP2 Also the distance vector ΔP
1is parallel to 119884-axis
(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP
1is not parallel to ΔP
2as shown
Z
YX
O
m2m
m + 2m + 1
1
23
42m minus 1
m minus 1
⋱⋱
Figure 7 The structure of spherical conformal array
in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution
For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation
9 Simulation Results
In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896
1120579= 05
1198961120593
= 05 1198962120579
= 03 1198962120593
= 07 The sensor patterns are119892119894120579= sin(120579
119895minus 120593
119895) and 119892
119894120593= cos(120579
119895minus 120593
119895) in the global
coordinate 120579119895and 120593
119895are the elevation and azimuth of the
119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as
RMSE = radic 1
500
500
sum
119905=1
[(120579119894119905minus 120579)
2
+ (120593119894119905minus 120593)
2
] (70)
where 120579119894119905and 120593
119894119905are the elevation and azimuth estimators of
the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)
For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]
In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 11
Y
XO
m
m minus 11
2
middot middot middot
middot middot middot
ΔP1
(a) The distance vector ΔP1
O
Y
X
m + 2
m + 1
ΔP2
(b) The distance vector ΔP2
Figure 8 The schematic diagram of the distance vector
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1ESPRIT source 2
minus10 minus5 0 5 10 15 20
(a) The success rate versus SNR
SNR (dB)
RMSE
(deg
)
PARAFAC source 1 PARAFAC source 2 ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
07
06
05
04
03
02
01
0minus10 minus5 0 10 15 20
(b) The RMSE versus SNR
Figure 9 The estimation performance versus SNR
experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices
are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance
We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 International Journal of Antennas and Propagation
100 200 300 400 500 600 700 800 900 100050
55
60
65
70
75
80
85
90
95
100
Number of snapshots
Succ
ess r
ate (
)
PARAFAC source 1 PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
(a) The success rate versus number of snapshots
100 200 300 400 500 600 700 800 900 10000
002
004
006
008
01
012
014
016
Number of snapshots
RMSE
(deg
)
PARAFAC source 1PARAFAC source 2ESPRIT source 1
ESPRIT source 2 CRB source 1 CRB source 2
(b) The RMSE versus number of snapshots
Figure 10 The estimation performance versus number of snapshots
0
10
20
30
40
50
60
70
80
90
100
SNR (dB)
Reso
lutio
n pr
obab
ility
()
PARAFAC source 1PARAFAC source 2
ESPRIT source 1 ESPRIT source 2
minus10 minus5 0 5 10 15 20
Figure 11 The resolution probability versus SNR
the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array
Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation
conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579
1minus 120579
1| |120579
2minus 120579
2| are
smaller than |1205791minus 120579
2|2 Meanwhile |120593
1minus 120593
1| |120593
2minus 120593
2| are
smaller than |1205931minus 120593
2|2 120579
119894and 120579
119894represent the estimated
and real elevations for 119894th incident signal respectively 120593119894
and 120593119894represent the estimated and real azimuths for 119894th
incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise
10 Conclusion
In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 13
Appendix
The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by
S ⊙ T = [s1otimes t
1 s
119875otimes t
119875] isin C
119862119863times119875 (A1)
where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by
s otimes t =[[[[
[
1199041t
1199042t119904119862t
]]]]
]
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)
References
[1] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010
[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013
[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012
[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013
[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009
[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011
[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999
[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008
[9] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011
[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011
[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005
[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013
[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012
[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012
[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007
[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013
[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012
[18] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011
[19] L Zou J Lasenby and Z S He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012
[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013
[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000
[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000
[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988
[24] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and Its Applications vol 18 no 2pp 95ndash138 1977
[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 International Journal of Antennas and Propagation
[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998
[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998
[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991
[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm
[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of