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Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Arrays: A Novel Approach to ArrayProcessing with Enhanced Degrees of Freedom
Xiangfeng Wang
OSPAC
May 7, 2013
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Reference
Reference
Pal Piya, and P. P. Vaidyanathan. Nested arrays: anovel approach to array processing with enhanced degrees offreedom. IEEE Transactions on Signal Processing.58.8(2010): 4167-4181.
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Outline
Outline
Motivation and Goal
Co-array Perspective
Nested Array: Optimization
DOA Estimation
Numerical Examples
Conclusion
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Motivation and Goal
Motivation and Goal
Antenna arrays
Direction-of-arrival (DOA) estimation
Uniform linear arrays (ULA)
Degree of freedom (DOF) is N − 1 for ULA
Goal: nested arrrays
obtain O(N2) DOF from only O(N) physical sensors
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Co-array Perspective
Signal Model
A N element possibly nonuniform linear antenna array
a(θ) ∈ RN×1 be the steering vector
a(θ)i = e j · 2πλ·di ·sin θ
where di denotes the position of the i-th sensor
Assume D narrowband sources from directions{θi , i = 1, 2, · · · ,D} with powers {σ2i , i = 1, 2, · · · ,D}Received signal
x[k] = A · s[k] + n[k]
where A = [a(θ1) a(θ2) · · · a(θD)] denotes the array manifoldmatrix and s[k] = [s1[k] s2[k] · · · sD [k]]T , and n[k] denotestemporally and spatially white, and uncorrelated from thesources.
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Co-array Perspective
Difference Co-Array
Rxx = E[xxH
]= ARssA
H + σ2nI
= A
σ21
σ22...σ2D
AH + σ2nI
z = vec(Rxx) = vec
[D∑i=1
σ2i (a(θi )aH(θi ))
]+ σ2n1n
= (A∗ � A) p + σ2n1n
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Co-array Perspective
Difference Co-Array
A∗ � A behave like the manifold of a array whose sensorlocations are given by the distinct values in the set{xi − xj , 1 ≤ i , j ≤ N}Difference Co-Array: Let us consider an array of N sensors,with xi denoting the position vector of the i-th sensor. Definethe set
D = {xi − xj}, ∀i , j = 1, 2, ·,N,
where Du denotes the distinct elements of the set D.
The difference co-array is defined as the array which hassensors located at positions given by the set Du.
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Co-array Perspective
Difference Co-Array
Weight Function: Define an integer valued functionw : Du 7→ N+ such that w(d) = no. of occurences of d in D,d ∈ Du.
w(0) = N, 1 ≤ w(d) ≤ N − 1,∀d ∈ Du\{0}.w(d) = w(−d),
∑d∈Du ,d6=0 w(d) = N(N − 1)
Cardinality of Du gives the defrees of freedom
DOFmax = N(N − 1) + 1
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Co-array Perspective
Difference Co-Array
Summary: If we use second-order statistics, then by exploitingthe degrees of freedom (DOF) of the difference co-array, thereis a possibility that we can get O(N2) degrees of freedomusing only O(N) physical elements
Examples:
an N element ULA ⇒ 2N − 1 elements ULA
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
Nested Array
Idea: Use a possibly nonuniform array ⇒ its differenceco-array has significantly more degrees of freedom than theoriginal array.
Nested Array: can be generated very easily in a systematicfashion and degrees of freedom of its co-array can be exactlypredict.
Two Level Nested Array
K Levels Nested Array
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
Two Level Nested Array
A concatenation of two ULAs
Inner ULA has N1 elements with spacing d1
Sinner = {md1,m = 1, 2, · · · ,N1}
Outer ULA has N2 elements with spacing d2 = (N1 + 1)d1
Souter = {n(N1 + 1)d1, n = 1, 2, · · · ,N2}
Difference co-array: a ULA with 2N2(N1 + 1)− 1 elements
Sca = {nd1, n = −M, · · · ,M,M = N2(N1 + 1)− 1}
2N2(N1 + 1)− 1 freedoms in the co-array using only N1 + N2
elements
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
Two Level Nested Array
N1 = 3, d1 = d , N2 = 3, d2 = 4d , M = 11
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
Two Level Nested Array
maxN1,N2
2N2(N1 + 1)− 1
s.t. N1 + N2 = N
⇓
N optimal N1, N2 DOF
even N1 = N2 = 12N N2−2
2+ N
odd N1 = N−12
, N2 = N+12
N2−12
+ N
obtain little over half of the maximum
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
K Levels Nested Array
Parameters: K , Ni , i = 1, 2, · · · ,K ∈ N+
S1 = {nd , n = 1, 2, · · · ,N1}
Si = {nd ·i−1∏j=1
(Nj + 1), n = 1, 2, · · · ,Ni}, i = 2, · · · ,K
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
K Levels Nested Array
Degrees of freedom in the corresponding difference co-array
DOFK = 2 {[N2(N1 + 1)− 1] + [(N3 − 1)(N1 + N2 + 1)
+(N1 + 1)] + · · ·+ [(NK − 1)(N1 + N2 + N3 + · · ·· · ·+ NK−1 + 1) + (N1 + N2 + · · ·+ NK−2 + 1)}+ 1
⇓
DOFK = 2
K∑i=1
K∑j=i+1
NiNj + NK − 1
+ 1
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
K Levels Nested Array
maxK∈N+
maxN1,··· ,NK∈N+
DOFK
subject toK∑i=1
Ni = N
Theorem 1. Given a number N of sensors, the optimal number ofnesting levels K and the number of sensors per nesting level aregiven by
K = N − 1,
Ni =
{1, i = 1, 2, · · · ,K − 1,0, i = K .
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
K Levels Nested Array
DOFK = 2
1
2
( K∑i=1
Ni
)2
−K∑i=1
N2i
+ NK
− 1
= N2 −K∑i=1
N2i + 2NK − 1
1 ≤ j ≤ K − 1 ⇒ break Nj into the sum of two smallerintegers Nj1 and Nj2, i.e., Nj = Nj1 + Nj2 ⇒ (K + 1) levels ofnested with {N1,N2, · · · ,Nj−1,Nj1,Nj2,Nj+1, · · · ,NK}sensors ⇒
DOFK+1 = N2 −
∑i 6=j ,i=1
N2i + N2
j1 + N2j1
+ 2NK − 1
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Nested Array
K Levels Nested Array
∆DOF = DOFK+1 −DOFK = N2j −N2
j1 −N2j2 = 2Nj1Nj2 ≥ 0
breaking up always increases the degrees of freedom
similar analysis for j = K
∆DOF = DOFK+1 − DOFK = 2NK1(NK2 − 1) ≥ 0 �
The corresponding difference array is a nonuniform linear arraywith degrees of freedom given by DOFopt = N(N − 1) + 1which is same as the upper bound.
Structure of the Optimally Nested Array: the optimum nestedarray has sensors located at the positions given by the setSopt = {d , 2d , 4d , 8d , · · · , 2N−1d}The optimally nested array is one with exponential spacing
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Application
Spatial Smoothing Based DOA Estimation
Spatial smoothing works only for a ULA and we shall focus onthe two-level nested array or any array whose differenceco-array is a filled ULA.
2-level nested array, N2 sensors in each level
A∗ � A ∈ RN2×D with (N2 − 2)/2 + N distinct rows
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Application
Spatial Smoothing Based DOA Estimation
Theorem 2 : The matrix Rss as defined as
Rss = R̂2
where
R̂ =1√
N2
4 + N2
(A11ΛAH11 + σ2nI )
has the same form as the covariance matrix of the signalreceived by alonger ULA consisting of N2/4 + N/2 sensorsand hence by applying MUSIC on Rss, uptp N2/4 + N/2− 1sources can be identified.
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Numerical Examples
Environment
6 sensor array (N = 6)
8 narrowband sources (D = 8)
directions of arrival {−60◦,−45◦,−30◦, 0◦, 15◦, 30◦, 45◦, 60◦}noise is assumed to be spatially and temporally white
2 level nested array with 3 sensors in each level
N2/4 + N/2− 1 = 11
SS-method: proposed spatial smoothing based technique
QS-method: KR product based MUSIC in [1] which requiresquasi stationarity
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Numerical Examples
MUSIC Spectrum with different numbers of snapshots
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Numerical Examples
MUSIC Spectrum with different numbers of snapshots
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Numerical Examples
Optimally Nested Array vs 2 Level Nested Array
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Numerical Examples
Optimally Nested Array vs 2 Level Nested Array
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Conclusion
Conclusion
A novel nested array structure is proposed which can realizesignificantly more degrees of freedom
Optimum nested array structure was found through solving acombinatorial optimization problem
An alternative spatial smoothing based approach tounderdetermined DOA estimation
· · ·Future research ?
Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom
Conclusion
References
W.K. Ma, T.H. Hsieh, and C.Y. Chi. DOA estimation ofquasi-stationary signals via Khatri-Rao subspace. ICASSP,April 2009: 2165–2168.