CFAR Detection In Clutter With Unknown Correlation Properties

CFAR Detection In ClutterWith Unknown CorrelationProperties

R. S. RAGHAVAN, Member, IEEE

H. F. QIU

D. J. McLAUGHLIN, Member IEEENortheastern University

We develop a constant false-alarm rate (CFAR) approach fordetecting a random N-dimensional complex vector in the presenceof clutter or interference modeled as a zero-mean complexGaussian vector whose correlation properties are not known to thereceiver. It is assumed that estimates of the correlation propertiesof the clutter/interference may be obtained independently byprocessing the received vectors from a set of reference cells. Wecharacterize the detection performance of this algorithm whenthe signal to be detected is modeled as a zero-mean complexGaussian random vector with unknown correlation matrix. Resultsshow that for a prescribed false alarm probability and a givensignal-to-clutter ratio (to be defined in the text), the detectabilityof Gaussian random signals depends on the eigenvalues of thematrix R¡1c Rs. The nonsingular matrix Rc and the matrix Rs arethe correlation matrices of clutter-plus-noise and signal vectorsrespectively. It is shown that the “effective” fluctuation statisticsof the signal to be detected is determined completely by theeigenvalues of the matrix R¡1c Rs. For example, the signal to bedetected has an effective Swerling II fluctuation statistics when alleigenvalues of the above matrix are equal. Swerling I fluctuationstatistics results effectively when all eigenvalues except one areequal to zero. Eigenvalue distributions between these two limitingcases correspond to fluctuation statistics that lie between SwerlingI and II models.

Manuscript received April 6, 1993; revised February 9 and March 25,1994.

IEEE Log No. T-AES/31/2/09744.

This work was supported by the U.S. Air Force Rome Laboratoryunder contracts F19628-91-K-0047 and F19628-93-K-0005.

Authors’ address: Dept. of Electrical and Computer Engineering,Northeastern University, 409 Dana Bldg., Boston, MA 02115.

0018-9251/95/$10.00 c° 1995 IEEE

I. INTRODUCTION

There is a need to consider the performance ofvarious constant false alarm rate (CFAR) algorithmswhich coherently process received signals that aremulti-dimensional in nature. In the CFAR context,results of such analysis may be useful in assessingthe potential benefits of utilizing the capabilityof radar systems that can acquire and processmulti-dimensional (or vector) signals. For example,the various components of the received vector mayrepresent 1) a time series of the scattered signalfor a given resolution cell, or 2) scattered signalsfor different transmit and/or receive polarizationsof the radar antennae, or 3) scattered signals fromdifferent subbands of a transmitted wideband signal, or4) an arbitrary combination of the above. The CFARrequirement of a candidate algorithm ensures that thefalse alarm rate may be prescribed at a given valueindependent of the correlation properties betweenthe various components of clutter. Thus, much likethe well known scalar CFAR problem, vector CFARalgorithms may be developed if one presupposes thatthe unknown correlations between the various cluttercomponents may be implicitly estimated from radarreturns obtained for a set of adjacent reference cells(or secondary cells) which are assumed to have thesame correlation properties as the clutter in the testcell (or primary cell).A number of authors [1—3] have addressed the

problem of detecting a known signal (known to withina complex multiplicative factor) in interference ornoise modeled as a zero-mean complex GaussianN-vector whose correlation properties are notknown. The work described in this paper assumesthat the signal to be detected is either an unknowndeterministic vector, or is a realization of a zero-meancomplex Gaussian N-vector. In the latter case, thecorrelation matrix of the signal to be detected isunknown to the receiver.The CFAR algorithm is described in Section II,

where expessions for the detection probability PD andthe false alarm probability PFA are given. Section IIIdescribes some results. Summary and conclusionsare given in Section IV. Material of a supplementarynature are included in the Appendices.

II. CFAR ALGORITHM AND EVALUATION OF ITSPERFORMANCE

In [1] Kelly has developed the generalizedlikelihood ratio (GLR) test for detecting a known(known to within a complex multiplicative constant)signal in the presence of noise/interference modeledas a zero-mean complex Gaussian N-vector. A setof secondary N-vectors z(k); k = 1,2, : : :K, (K ¸N)which are mutually independent and have the samestatistical properties as the interference in the test cell

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995 647

(or primary cell) is assumed to be given. The resultingtest is shown to be CFAR (i.e., the probability of falsealarm is independent of the correlation matrix of theadditive noise/interference). If one assumes at theoutset that the signal to be detected is an unknowndeterministic complex N-vector, a uniformly mostpowerful (UMP) test does not exist for detectingthe presence of such a signal in clutter/interference.However, as shown in Appendix A, a test that isinvariant to certain transformations of the data andwhich is UMP for this restricted class (called UMPinvariant test [4—6]) may be developed using the GLRformulation. In this context, we seek a test that isinvariant to any linear transformation of the receivedvectors by a nonsingular matrix. As shown in AppendixA the GLR test for the case when the signal is anunknown deterministic vector is given by the quadraticform (denoted by the function Á for convenience):

Á(z,S)´ zS¡1z?(y0¡ 1) (1)

where z is the complex N-vector from the primaryor test cell and is given by the sum of the signal andclutter under hypothesis H1. Under hypothesis H0,z is made up of only clutter. We note that thermalnoise is always present in the received vector forboth hypotheses and we use the term clutter toimply clutter-plus-noise throughout this work. TheHermitian matrix S is proportional to the maximumlikelihood estimate of the clutter correlation matrixRc as obtained from the K secondary vectors z(i);i= 1,2, : : :K and is given by

S =KXi=1

z(i)z(i) (2)

in (1) and (2) (and throughout this work) the symbolis used to denote the conjugate transpose of a vector.Finally in (1) we find it convenient to express thedetection threshold as (y0¡ 1) as in [1]. Obviouslyy0 > 1 and as shown below, y0 is determined from therequired probability of false alarm.The statistic Á(z,S) is invariant to any

transformation of the data by a nonsingular matrixC. Thus, with z = Cz and S = CSC we observe thatÁ(z,S) = Á(z,S). In Appendix A we show that thefunction Á assigns a unique value to the set of allcomplex N-vectors z and Hermitian positive definitematrices S generated by the transformations: z! Czand S! CSC , where C is any nonsingular matrix.The properties of uniqueness and invariance definethe function Á as a maximal invariant [4, 5] and thequadratic Á(z,S) as a maximal invariant statistic forthe given problem. The maximal invariant plays acentral role in invariant hypothesis testing problemssince all invariant functions (invariant to nonsingulartransformations of the data in the present context) canbe expressed as functions of the maximal invariant.

This means that all invariant tests may be expressedas functions of the maximal invariant statistic andone does not need anything else from the data forsolving a given hypothesis testing problem subjectto the invariance restriction. In Appendix A, theoptimality of the test in (1) subject to the invariancerestriction is established by showing that the maximalinvariant statistic has a monotone likelihood ratio (see[5, pp. 135] for example).The invariance requirement has some useful

consequences from a practical viewpoint. Inpolarimetric detection for example, it is unimportantwhether the orthogonal polarizations used arevertical (V) and horizontal (H); right-hand circular(RHC) and left-hand circular (LHC). In fact anytwo nonidentical polarizations may be used as basisvectors. Furthermore, whatever the polarization basisvectors, the receiver does not need this information forprocessing because of the invariance property of thetest employed.The probability density function (pdf) of the

detection statistic in (1) for both hypotheses H0 andH1 (when the signal to be detected is a deterministicvector) is well known in the technical literature formultivariate analysis as the central F-distribution andthe noncentral F-distribution, respectively (see [7] forexample). However, we use results in [1] as a startingpoint so that expressions for the detection probabilitydo not become too cumbersome to evaluate for therandom signal case.We first obtain the expression for the false alarm

probability (PFA). Let the random variable Y denotethe quantity ZS¡1Z. The pdf of Y given hypothesis H0is given by (see [7] or [1, Appendix] for example):

fY(y=H0) =K!

(K ¡N)!(N ¡1)!yN¡1

(1+ y)K+1U(y) (3)

where U(y) is the unit step function and K is thenumber of secondary vectors (or reference vectors)used for estimating the covariance matrix in (2).The probability of false alarm is obtained by

integrating (3) and may be shown (see Appendix B)to be given by

PFA =μ

K

N ¡ 1

¶μ1y0

¶K¡N+1£ 2F1(K ¡N +1,1¡N;K ¡N +2;y¡10 ) (4)

where 2F1(®,¯;°;y) is the Gauss hypergeometric seriesgiven by

2F1(®,¯;°;y) = 1+1Xk=1

(®)k(¯)k(°)k

yk

k!(5)

where

(μ)k =k¡1Yi=0

(μ+ i); k ¸ 1: (6)

648 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

Since the second argument of the GaussHypergeometric series is negative (for N > 1), thequantity (1¡N)k is zero for all k ¸N and so onlya finite number of terms (=N) need to be summedto obtain the PFA in (4). Expression (4) is also usefulfor obtaining the appropriate thereshold (y0¡ 1) fora prescribed false alarm probability. We also notethat the PFA expression does not involve the cluttercorrelation matrix Rc either implicitly or explicitly.This confirms the CFAR property of the detectionalgorithm.Although the GLR test in (1) was developed for

detecting an unknown deterministic signal in clutter,it may still be used for detecting random signals(including non-Gaussian random signals) since thetest is UMP invariant. That is, there exists no otherinvariant test that provides a higher probability ofdetection (for each realization of the random signal)and for a given PFA than the test in (1). Thereforesubject to the invariance requirement, the test isoptimum–independent of the pdf of the signal vector.We wish to evaluate the detection performance ofthe test when the signal is modeled as a zero-meancomplex Gaussian N-vector. To obtain an expressionfor the detection probability, we set the test cellvector Z = Zc+Zs, where the N-vector Zc is clutter orinterference only and the N-vector Zs is the randomsignal vector which is to be detected. Using theexpression for detection probability given in [1] weshow in Appendix C that for a given value of therandom variable X = ZsR

¡1c Zs, the probability of

detection for the test in (1) may be expressed as

P[Y > (y0¡ 1)=H1,X = x]

= 1¡ 1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶£ (y0¡ 1)N+i¡1Gi(x=y0) (7)

where Gi(x) is related to the incomplete Gammafunction ¡ (i,x) by

Gi(x) = e¡x

i¡1Xn=0

xn

n!=¡ (i,x)(i¡ 1)!: (8)

The implicit assumption that the correlation matrix Rcis nonsingular is generally observed in practice sincethe presence of receiver thermal noise modeled as anadditive white noise process precludes the completesuppression of clutter-plus-noise.And so the overall probability of detection may

be determined if the pdf of the random variableX = ZsR

¡1c Zs is known. We consider three separate

cases below.

1) Zs = zs, where zs is an unknown deterministicvector.

Defining the signal-to-clutter ratio ¸0 = E[X], wehave in this case:

fX(x) = ±(x¡¸0): (9)

And PD is obtained by averaging (7) over the density in(9) to obtain

PD = 1¡1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶£ (y0¡ 1)N+i¡1Gi(¸0=y0): (10)

2) Zs = ®zs, where zs is an unknown deterministicvector and ® is a zero-mean complex Gaussian randomvariable.In this case the random variable X has an

exponential density given by

fX(x) =e¡x=¸0

¸0U(x) (11)

where the signal-to-clutter ratio is again ¸0 = E[X] =E[j®j2]zsR¡1c zs.Substituting (11) in (7) and integrating with respect

to x the detection probability is given by

PD = 1¡1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶

£ (y0¡ 1)N+i¡1"1¡μ

¸0=y01+¸0=y0

¶i#: (12)

In obtaining (12) we have used the expression:Z 1

0Gi(x=y0)fX(x)dx= 1¡

μ¸0=y0

1+¸0=y0

¶i: (13)

3) Zs is an N-dimensional zero-mean complexGaussian random vector whose correlation matrix(unknown to the receiver) is Rs.To obtain the pdf of X = ZsR

¡1c Zs we observe

that the N-vector Zs is statistically equivalent to thevector R1=2s Ys, where the N-vector Ys is a complex,zero-mean, white Gaussian vector with E[YsYs] = Iand R1=2s is the square root of the matrix Rs. We couldhave more easily defined Ys = R

¡1=2s Zs, but the above

method of reasoning avoids the necessity for treatingthe case when Rs is a singular matrix as a separatecase. Thus X is statistically equivalent to the innerproduct of the vector R¡1=2c R

1=2s Ys with itself, i.e.,

X ´ fR¡1=2c R1=2s YsgfR¡1=2c R

1=2s Ysg= YsR1=2s R¡1c R

1=2s Ys.

Next, the Hermitian matrix R1=2s R¡1c R1=2s is written

in the form R1=2s R¡1c R

1=2s =Q¤Q, where Q is a

unitary matrix whose columns are the eigenvectors ofR1=2s R¡1c R

1=2s and ¤= diag[¸1,¸2, : : : ,¸N] is a diagonal

matrix containing the eigenvalues of R1=2s R¡1c R1=2s . But

given two N-by-N matrices A and B, we know thatthe eigenvalues of matrix AB are identical to those

RAGHAVAN ET AL.: CFAR DETECTION IN CLUTTER WITH UNKNOWN CORRELATION PROPERTIES 649

Abbas

Highlight

of matrix BA (see [8] for example). Therefore, theeigenvalues of R1=2s R¡1c R

1=2s are identical to those

of R¡1c Rs (which are also equal to the eigenvaluesof the matrix RsR

¡1c ). Finally, we define a complex

N-vector Ys as Ys =QYs and since Q is a unitary matrix,the N components of Ys are zero-mean mutuallyindependent complex Gaussian random variables andthe correlation between the ith and jth componentsis given by E[YsiY

¤sj] = ±i,j, where ±i,j is the Kronecker

delta. Thus the random variable X is statisticallyequivalent to

X ´NXi=1

¸iXi (14)

where f¸i; i= 1,2, : : :Ng are the eigenvalues of R¡1c Rs(the sum on the right hand side will involve less thanN terms if some of the eigenvalues are zero as is thecase when Rs is a singular matrix). Xi, i= 1,2, : : :N,are independent exponentially distributed randomvariables with E[Xi] = 1, i= 1,2, : : :N.

The pdf of X may be obtained from its momentgenerating function (MGF) which is given by

©X(s) =Z 1

0fX(x)e

¡sx dx=NYi=1

1(1+¸is)

: (15)

Thus,

fX(x) =12¼j

Z c+j1

c¡j1©X(s)e

sx ds, x¸ 0:(16)

The contour c lies to the right of all singularities of©X(s) in the left-half s-plane.Substituting (15) and (16) in (7) and integrating

with respect to the variable x we get:

PD = 1¡1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶(y0¡1)N+i¡1

£"i¡1Xk=0

12¼j

Z c+j1

c¡j1

©X(s)y0(1¡ y0s)k+1

ds

#: (17)

To obtain (17) we make use of the followingintermediate result:Z 1

0Gi(x=y0)e

sx dx=i¡1Xk=0

y0(1¡ y0s)k+1

: (18)

Finally, the contour integral in (17) is evaluated byobtaining the residue of the integrand at each polethat lies inside the contour and adding all the residues.Note that the multiple pole at s= 1=y0 lies in theright-half s-plane and is therefore not enclosed by thecontour. The remaining poles are all located inside thecontour since the non-zero eigenvalues of R¡1c Rs areall positive. Results are given for the case where all the

poles of the integrand (at sl =¡1=¸l; l = 1,2, : : :N aredistinct; other cases may be similarly handled. And so,

12¼j

Z c+j1

c¡j1

©X(s)y0(1¡ y0s)k+1

ds

=NXl=1

(y0=¸l)(1+ y0=¸l)k+1

264 NYm=1m 6=l

1(1¡¸m=¸l)

375 :(19)

Substituting (19) into (17) we obtain after somesimplification:

PD = 1¡1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶(y0¡ 1)N+i¡1

£NXl=1

[1¡ (1+ y0=¸l)¡i]NYm=1m 6=l

1(1¡¸m=¸l)

:

(20)

The signal-to-clutter ratio for this case is defined as¸0 = E[X] =

PNi=1¸i. This is also equal to the trace

of the matrix R¡1c Rs. Unlike the two earlier casesdiscussed in this section, the detection probability(for given K, N, and PFA) in (20) is not a function ofthe single parameter ¸0 (= signal-to-clutter ratio).PD is determined by the precise manner in which theeigenvalues of R¡1c Rs are distributed. Two limitingcases are of interest here.First, consider the case where all eigenvalues

of R¡1c Rs except one are equal to zero (theconclusions reached here also apply to situationswhere one eigenvalue is large compared with theremaining). Without loss of generality let ¸1 be thenon-zero eigenvalue. Thus the random variable X isexponentially distributed and the signal-to-clutter ratiois E[X] = ¸1. The detection probability in this case isthe same as that given in (12). As far as the receiveris concerned, the signal to be detected appears tohave an effective Swerling I fluctuation (scan-to-scanamplitude fluctuation). These signals require a largersignal-to-clutter ratio to achieve a PD close to 1 thanthe Swerling II case [9] (since there is a greater chancefor all N components of the signal to be in a fade).Next, consider the case where all eigenvalues of

R¡1c Rs are equal (for example consider Rs = ¸Rc). Theconclusions reached here also apply to situations wherethe eigenvalues are all not necessarily equal but aretightly clustered together in a single group. Hence ¸i =¸0=N; i= 1,2, : : :N. The pdf of the random variable Xis the chi-square density with 2N degrees of freedomand the signal-to-clutter ratio is E[X] = ¸0. Thus, allN components of the received vector appear to fadeindependently, i.e., in effect, the signal appears to havethe Swerling II fluctuation statistics (pulse-to-pulseamplitude fluctuation).


The pdf of X in this case is

fX(x) =xN¡1

(¸0=N)N(N ¡ 1)!e¡xN=¸0U(x): (21)

The probability of detection is obtained by averaging(7) over the pdf in (21) to obtain

PD = 1¡1

yK0 [1+ (¸0=Ny0)]N

£K¡N+1Xi=1

μK

N + i¡ 1

¶(y0¡ 1)N+i¡1

£i¡1Xk=0

μN ¡ 1+ k

k

¶·(¸0=Ny0)

1+ (¸0=Ny0)

¸k: (22)

It is of interest to note that in all casesdiscussed above, the limit of PD as ¸0! 0 (i.e., zerosignal-to-clutter ratio) is given by

lim¸0!0

PD = 1¡1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶(y0¡ 1)N+i¡1:

(23)

If the term inside the sum is expanded in a binomialseries and the resulting expression simplified, (23) maybe shown to be identical to the PFA expression given in(4) as expected.

III. RESULTS

A few sample results for the various casesdiscussed in Section II are shown here for illustrativepurposes. In Fig. 1, we compare the detectionperformance of the algorithm in (1) with the GLRalgorithm developed in [1]. In [1] the signal to bedetected zs is assumed to be known a priori (exceptfor an unknown complex multiplicative constant). TheGLR test provided here for completeness is [1]

j(zsS¡1z)j2(zsS¡1zs)(1+ zS¡1z)

? (y0¡ 1)y0

: (24)

The threshold in this case is chosen such that PFA =1=yK¡N+10 .In Fig. 1(a) we set K = 10 and N = 9, while in

Fig. 1(b) K = 10 and N = 5 (note that the scaleon the abscissa is different in the two figures). Thethreshold y0¡ 1 in each case is selected such thatPFA = 10

¡4. Note that the signal to be detected in boththese figures is deterministic. In each case simulationresults (indicated by the symbols “O” and “X”) arealso shown to provide an independent confirmationof the analytical results. A total of 103 independenttrials were used to obtain estimates of the detectionprobability. Note that the number of trials requireddoes not need to be several orders of magnitudegreater than 1=PFA since the false alarm performance

Fig. 1(a). Probability of detection for deterministic signal versussignal-to-clutter ratio (in dB) using tests specified in (1) and (24).Parameters used are K = 10, N = 9, PFA = 10

¡4. Lines indicateanalytical results and symbols indicate results obtained from Monte

Carlo simulations.

Fig. 1(b). Probability of detection for deterministic signal versussignal-to-clutter ratio (in dB) using tests specified in (1) and (24).Parameters used are K = 10, N = 5, PFA = 10

¡4. Lines indicateanalytical results and symbols indicate results obtained from Monte

Carlo simulations.

is not being tested in these simulations. As seen inthese figures, the detection loss of the test specifiedin expression (1) is approximately 2 dB with respectto the test in (24) for the parameters used in theseexamples.Fig. 2 shows results for the two limiting cases

encountered when the signal to be detected is azero-mean complex Gaussian random vector. In Fig. 2,“case 1” represents the situation where all eigenvaluesof the matrix R¡1c Rs except one are equal to zero.As described earlier, these results also describe thesituation where the signal to be detected is expressibleas Zs = ®zs (zs is a deterministic signal not knowna priori to the receiver, and ® is a zero-mean complexGaussian random variable). The detection performancewhen all eigenvalues of R¡1c Rs are equal is marked“case 2”. Both analytical and simulation results(simulations marked by symbols O and X) are shownfor the same set of parameters as in Fig. 1. Thenumber of independent trials used in these simulationswas 103. For reasons given in the previous section,case 1 represents the most conservative estimate of the


Fig. 2(a). Probability of detection for random signal versussignal-to-clutter ratio (in dB) using test specified in (1). Parameters

used are K = 10, N = 9, PFA = 10¡4. Case 1 indicates all

eigenvalues of R¡1c Rs except one are zero and case 2 indicates alleigenvalues of R¡1c Rs are equal. Lines indicate analytical resultsobtained from expressions (12) and (22) while symbols indicate

results obtained from Monte Carlo simulations.

Fig. 2(b). Probability of detection for random signal versussignal-to-clutter ratio (in dB) using test specified in (1). Parameters

used are K = 10, N = 5, PFA = 10¡4. Case 1 indicates all

eigenvalues of R¡1c Rs except one are zero and case 2 indicates alleigenvalues of R¡1c Rs are equal. Lines indicate analytical resultsobtained from expressions (12) and (22) while symbols indicate

results obtained from Monte Carlo simulations.

detection performance (for high signal-to-clutter ratios)given that the signal to be detected is a zero-meancomplex Gaussian vector. Similarly, case 2 representsthe best achievable performance of the algorithm(again for high signal-to-clutter ratios) given thatthe signal to be detected is a zero-mean complexGaussian vector. The detection performance for otherdistributions of the eigenvalues lie between the twocases shown in these figures.

IV. SUMMARY AND CONCLUSIONS

We have considered a CFAR approach fordetecting a complex random N-vector in a backgroundof homogeneous clutter. The detection statistic isthe quadratic form ZS¡1Z, where the matrix S isproportional to the maximum likelihood estimate ofthe clutter correlation matrix Rc. S is obtained by

processing the secondary (or reference) vectors. Thetest was shown to be UMP invariant and detectionperformance of the algorithm was characterized whenit is used for detecting signals that are zero-meanGaussian random vectors. An expression forthe probability of false alarm is given in (4) andexpressions for the probability of detection weredeveloped for various assumed statistics of the signalto be detected. It was shown that for a prescribedfalse alarm probability and a given signal-to-clutterratio the detectability of Gaussian random signalsdepends on the eigenvalues of the matrix R¡1c Rs.Further, the “effective” fluctuation statistics of thesignal to be detected is determined completely bythese eigenvalues. As a possible application of thiswork, the expressions for PD may be used in systemcalculations in order to assess the potential benefitsof using certain types of multi-dimensional data toenhance the detection capability of CFAR systems.

APPENDIX A

In this Appendix, we develop the GLR test fordetecting an unknown deterministic complex N-vectorin clutter modeled as a zero-mean complex Gaussianrandom vector. We are concise in this developmentsince it parallels the treatment in [1]. Invariance ofthe test statistic to transformations of the data byany nonsingular matrix is indicated. Finally, the teststatistic is shown to be a maximal invariant statisticwith monotone likelihood ratio. These results showthat the test specified in (1) is UMP among the classof all invariant tests (see [5, pp. 135] for example).This Appendix is included primarily for completenessand much of these results especially for real vectorsand real matrices may be found in books concerningmultivariate statistical analysis. The results as shownhere are easily extended to the case where the vectorsare complex.The joint density function of the primary and

secondary cell vectors under the clutter-only hypothesis(H0) is written as

f(z(1),z(2), : : : ,z(K),z=Rc,H0)

=1

(¼N jRcj)K+1exp

"¡

KXi=1

z(i)R¡1c z(i)

#£ exp[¡zR¡1c z] (25)

where jRcj is the determinant of Rc. Since thequadratic form zR¡1c z is also equal to tr[R

¡1c zz] (tr[A]

is the trace of matrix A), the following estimate [1]of the clutter correlation matrix Rc0 maximizes thelikelihood function in (25):

Rc0 =1

K +1

Ãzz +

KXi=1

z(i)z(i)

!: (26)


We find it necessary to introduce the second subscript0 to remind us that this estimate is conditionedon hypothesis H0. Although the assumption K ¸N ¡ 1 is sufficient to guarantee that the matrix Rc0is nonsingular with probability 1, we see that thecondition K ¸N is required to make the estimateRc1 under hypothesis H1 to be nonsingular as well.Substituting the estimate Rc0 from (26) for Rc in (25),the maximum value of the likelihood function forhypothesis H0 is

f(z(1),z(2), : : : ,z(K),z=Rc = Rc0,H0)

=1

((e¼)N jRc0j)K+1: (27)

Similarly, the joint density function of the primaryand secondary cell vectors conditioned on hypothesisH1 and for given Rc and signal vector zs is written as

f(z(1),z(2), : : : ,z(K),z=zs,Rc,H1)

=1

(¼N jRcj)K+1exp

"¡

KXi=1

z(i)R¡1c z(i)

#

£ exp[¡(z¡ zs)R¡1c (z¡ zs)]: (28)

The choice zs = z in (28) maximizes the likelihoodfunction with respect to the signal vector zs. Theresulting expression may then be maximized withrespect to Rc by setting:

Rc1 =1

K +1

ÃKXi=1

z(i)z(i)

!: (29)

Substituting the estimates zs and Rc1 in (28),the maximum value of the likelihood function forhypothesis H1 is

f(z(1),z(2), : : : ,z(K),z=zs = zs,Rc = Rc1,H1)

=1

((e¼)N jRc1j)K+1: (30)

The GLR is obtained by taking the ratio of (30) and(27):

l10 =μ jS+ zz j

jSj¶K+1

(31)

where S is the sample correlation matrix defined in (2).The GLR test compares the quantity l10 in (31) to apreset threshold. Since jS+ zz j= jSj(1+ zS¡1z), theGLR test is eqivalent to comparing the statistic zS¡1zto a different threshold which is denoted by (y0¡ 1).Thus the GLR test is

Á(z,S)´ zS¡1z?(y0¡ 1): (32)

We use the notation Á(z,S) to denote the quadratic in(32) for convenience. Invariance of the statistic Á(z,S)

to transformations z = Cz and S = CSC where C isany nonsingular matrix is easily verified. We observethat Á(z,S) = zS

¡1z = z(C¡1C)S¡1(C¡1C)z = zS¡1z =

Á(z,S).To show that the function Á is a maximal invariant,

we need to show that Á assigns a unique value to theset of all complex N-vectors z and Hermitian positivedefinite matrices S generated by the transformations:z! Cz and S! CSC , where C is any nonsingularmatrix. Therefore, if Á(z,S) = Á(z,S), we must showthat there exists a nonsingular matrix C such thatz = Cz and S = CSC . To show this, we first definetwo N-vectors v = S¡1=2z and v = S

¡1=2z where S¡1=2

and S¡1=2

are the square root of matrices S¡1 and S¡1,

respectively (since both S and S are positive definite,each matrix has an inverse). Thus, Á(z,S) = Á(z,S)implies kvk2 = vv = vv = kvk2. The vectors v and vmust therefore be related by a unitary transformation:v =Uv (U is an N-by-N unitary matrix). The lastresult is easily shown by defining two N-by-N unitarymatrices U1 and U2, the orthonormal rows of eachmatrix may be constructed such that the first row ofU1 is v=kvk and the first row of U2 is v=kvk. Thus,U1v = [kvk,0,0 ¢ ¢ ¢0] = [kvk,0,0 ¢ ¢ ¢0] =U2v. Therefore,v =Uv where U =U1U2. Finally, v =Uv implies

S¡1=2

z =US¡1=2z or z = Cz and S = CSC , whereC = S

1=2US¡1=2. Hence Á is a maximal invariant.

To establish the monotone property of thelikelihood ratio, we observe that the pdfs of therandom variable Y = ZS¡1Z conditioned onhypotheses H0 and H1 are given by [7]

fY(y=H0) =K!

(K ¡N)!(N ¡1)!yN¡1

(1+ y)K+1U(y)

(33)

and

fY(y=X = x,H1)

= e¡x1F1(K +1;N;xy=(1+ y))

£ K!(K ¡N)!(N ¡1)!

yN¡1

(1+ y)K+1U(y) (34)

where the parameter x= zsR¡1c zs and 1F1(K +1;N;

xy=(1+ y)) is the degenerate hypergeometric functiondefined as

1F1(K +1;N;xy=(1+ y))

=·1+

(K +1)N

μxy

1+ y

¶

+(K +1)(K +2)N(N +1)2!

μxy

1+ y

¶2+ ¢ ¢ ¢

¸: (35)


Thus the likelihood ratio is given by

fY(y=X = x,H1)fY(y=H0)

= e¡x·1+

(K +1)N

μxy

1+ y

¶

+(K +1)(K +2)N(N +1)2!

μxy

1+ y

¶2+ ¢ ¢ ¢

¸(36)

which is an increasing function of y=(1+ y) and hencealso an increasing function of y. Note that the densityfunctions of the statistic Y are parametrized by thesingle quantity X (X = 0 for hypothesis H0). Thus themost powerful test of hypothesis H0 (X = 0) againstthe alternative H1 (X > 0) at a specified probability offalse alarm (size of the test) is given by (1) accordingto the Neyman—Pearson criterion. Results of thisAppendix show that the test in (1) is UMP invariant.

APPENDIX B

We develop an expression for the probability offalse alarm PFA in this Appendix.Using (3), the probability of false alarm is given by

PFA =Z 1

y0¡1fY(y=H0)dy

=Z 1

y0¡1

K!(K ¡N)!(N ¡ 1)!

yN¡1

(1+ y)K+1dy: (37)

With 1=(1+ y) = x, we have

PFA =K!

(K ¡N)!(N ¡ 1)!Z 1=y0

0xK¡N(1¡ x)N¡1dx:

(38)

The integral above may be expressed in terms of theincomplete beta function as

PFA =K!

(K ¡N)!(N ¡ 1)!B1=y0(K ¡N +1,N)(39)

where Bx(p,q) is the incomplete beta function [10]defined as

Bx(p,q) =Z x

0tp¡1(1¡ t)q¡1dt: (40)

The incomplete beta function is also expressed interms of the Gauss hypergeometric series (defined in(5)) by the relation [10]:

Bx(p,q) =xp

p 2F1(p,1¡ q;p+1;x): (41)

Substituting (41) in (39) the desired expression for thefalse alarm probability in (4) is obtained.

An alternate expression for PFA is given in (23)however the expression given in (4) is well suited forcalculation by computer since each term in the seriesmay be obtained by multiplying the preceding termby a suitable quantity. For example, the quantity

2F1(K ¡N +1,1¡N;K ¡N +2;y¡10 ) may becomputed efficiently as follows (we set M =K ¡N +1 and ®= y¡10 for convenience):

2F1(M,1¡N;M +1;®)

= 1+M(1¡N)(M +1)

®

+M(M +1)(1¡N)(2¡N)

(M +1)(M +2)®2

2!+ ¢ ¢ ¢

+M(M +1) ¢ ¢ ¢ (M +N ¡ 1)(1¡N)(2¡N) ¢ ¢ ¢ (¡1)

(M +1)(M +2) ¢ ¢ ¢ (M +N)

£ ®N¡1

(N ¡ 1)!´ a0 + a1 + a2 + ¢ ¢ ¢+ aN¡1 (42)

which may be evaluated using the following recursion:

a0 = 1

a1 = a0M(1¡N)(M +1)

®

a2 = a1(M +1)(2¡N)

(M +2)®

2¢¢¢

ak+1 = ak(M + k)(1¡N + k)

(M + k+1)®

(k+1)

for k = 0,1,2, : : :N ¡ 2:

(43)

This procedure prevents numerical overflows sincefactorials are not computed directly.

APPENDIX C

We develop an expression for detection probabilityconditioned on a given value of X = ZsR

¡1c Zs in this

Appendix. We use results given in [1] as a startingpoint.We start by assuming that the signal to be

detected is known a priori, i.e., Zs = zs. Thiscondition is removed later on. As described in [1], anN-dimensional signal space may be chosen such thatthe first coordinate axis represents the known signalvector zs. The remaining N ¡ 1 coordinate directionsare mutually orthogonal to each other. Each N-vectorfrom the primary cell and secondary cells may thusbe partitioned into two components Z = (Z1,Z2)

t. Thefirst component (denoted by Z1,Z1(1),Z1(2) : : : ,Z1(K)being complex scalars while the (N ¡ 1)-dimensionalcomplex vectors Z2,Z2(1),Z2(2) : : : ,Z2(K) representthe components of the respective N-vectors in the


directions orthogonal to the signal direction. Thedetection statistic Y when all vectors and matrices arepartitioned in this manner is shown to be expressible[1] in the form:

Y = Y1,2+Y2 (44)

where,Y2 = Z2S

¡122 Z2: (45)

The (N ¡ 1)-by-(N ¡ 1) matrix S22 is

S22 =KXk=1

Z2(k)Z2(k) (46)

and

Y1,2 =jZ1¡ S12S¡122 Z2j2PK

i=1 jZ1(i)¡ S12S¡122 Z2(i)j2(47)

and the 1-by-(N ¡ 1) matrix S12 is

S12 =KXk=1

Z1(k)Z2(k): (48)

Finally a random variable R is defined as R =(1+Y2)

¡1. Clearly, the range of R is the interval [0,1].We use the following result from [1]:

P[RY1,2> (y0¡ 1)=H1,R = r,Zs = zs,X = x]

= 1¡ 1

yK¡N+10

K¡N+1Xk=1

μK ¡N +1

k

¶£ (y0¡1)kGk(xr=y0) (49)

where the random variable X is the signal-to-clutterratio defined by the quadratic form X = ZsR

¡1c Zs and

the function Gk(¢) is defined earlier in (8). Note thatthe conditional probability given in (49) is independentof the actual signal direction (i.e., zs) as long as thesignal-to-clutter ratio X is a given value, and so thesignal zs could have been a realization of a randomvector.From (44) and the definition of R we see that

Y = Y1,2+R¡1¡ 1. Since Y1,2 in (47) is a nonnegative

random variable, it follows that the probability thatY > (y0¡1) is unity if 0· R · y¡10 . Similarly, fory¡10 · R · 1, the probability of Y > (y0¡1) is theprobability of Y1,2+R

¡1¡1> (y0¡ 1). Multiplying bothsides by R and rearranging we get

P[Y > (y0¡ 1)]= P[RY1,2> R(y0¡ 1)¡ 1+R]= P[RY1,2> (Ry0¡ 1)]:

Thus,

P[Y > (y0¡ 1)=H1,R = r,X = x]= 1 for 0· r · y¡10= P[RY1,2> (ry0¡ 1)=H1,R = r,X = x]

for y¡10 · r · 1: (50)

The second probability on the right hand side above isobtained by substituting ry0 instead of y0 in expression(49).Thus,

P[Y > (y0¡ 1)=H1,X = x]

=Z 1=y0

0fR(r)dr+

Z 1

1=y0fR(r)

£"1¡ 1

(ry0)K¡N+1

K¡N+1Xk=1

μK ¡N +1

k

¶

£ (ry0¡ 1)kGk(x=y0)#dr (51)

which may be rearranged to yield

P[Y > (y0¡1)=H1,X = x]

=Z 1

0fR(r)dr¡

K¡N+1Xk=1

μK ¡N +1

k

¶Gk(x=y0)

£Z 1

1=y0fR(r)

(ry0¡ 1)k(ry0)K¡N+1

dr: (52)

The pdf of random variable R is given by (see [1,Appendix])

fR(r) =K!

(K ¡N +1)!(N ¡ 2)!rK¡N+1(1¡ r)N¡2

0· r · 1: (53)

The first integral in (52) is 1 and the second integral isgiven by

Z 1

1=y0

(ry0¡1)k(ry0)K¡N+1

fR(r)dr

=K!

(K ¡N +1)!(N ¡ 2)!1

yK¡N+10

£Z 1

1=y0(ry0¡1)k(1¡ r)N¡2dr: (54)

With a change of variables: p= (ry0¡ 1)=(y0¡ 1), theintegral in (54) is given by

K!(K ¡N +1)!(N ¡ 2)!

(y0¡ 1)N+k¡1yK0

Z 1

0

pk(1¡p)N¡2dp

=

μK

N + k¡ 1

¶(y0¡ 1)N+k¡1

yK0: (55)


Substituting the result from (55) into (52) the followingresult is obtained

P[Y > (y0¡ 1)=H1,X = x]

= 1¡ 1yK0

K¡N+1Xi=1

μK

N + i¡ 1

¶£ (y0¡ 1)N+i¡1Gi(x=y0): (56)

ACKNOWLEDGMENT

We thank S. Bose and A. O. Steinhardt for theircomments and for sending us preprints of their work.We also thank the AES editor for radar, Dr. GaryKrumpholz for many helpful comments and suggestionswhich have undoubtedly enhanced this work.

REFERENCES

[1] Kelly, E. J. (1986)An adaptive detection algorithm.IEEE Transactions on Aerospace and Electronic Systems,AES-22, 1 (Mar. 1986), 115—127.

[2] Khatri, C. G., and Rao, C. R. (1987)Effects of estimated noise covariance matrix in optimalsignal detection.IEEE Transactions on Acoustics Speech and SignalProcessing, ASSP-35, 5 (May 1987), 671—679.

[3] Nitzberg, R. (1984)Detection loss of the sample matrix inversion technique.IEEE Transactions on Aerospace and Electronic Systems,AES-20, 6 (Nov. 1984), 824—827.

[4] Muirhead, R. J. (1982)Aspects of Multivariate Statistical Theory.New York: Wiley, 1982, ch. 6.

[5] Scharf, L. L. (1991)Statistical Signal Processing: Detection, Estimation andTime Series Analysis.New York: Addison-Wesley, 1991, ch. 4.

[6] Bose, S., and Steinhardt, A. O. (1994)The optimum invariant test for adaptive detection of aweak signal in unknown noise.To be published.

[7] James, A. T. (1964)Distribution of matrix variates and latent roots derivedfrom normal samples.Annals of Mathematical Statistics, 35 (1964), 475—501.

[8] Kailath, T. (1980)Linear Systems.Englewood Cliffs, NJ: Prentice-Hall, 1980, Appendix,A.25, 658.

[9] DiFranco, J. V., and Rubin, W. L. (1980)Radar Detection.Dedham, MA: Artech House, 1980.

[10] Gradshteyn, I. S., and Ryzhik, I. M. (1980)Table of Integrals, Series, and Products.New York: Academic Press, 1980, 950.


R. S. Raghavan (S’81–M’84) received a B.Tech. degree from the Indian Instituteof Technology, Madras, India, the M.S. degree from the University of Maine,Orono, and a Ph.D. from the University of Massachusetts at Amherst, all inelectrical engineering.He joined the Department of Electrical and Computer Engineering at

Northeastern University, Boston, MA, in 1985, where he is now AssociateProfessor. His current research interests are in the area of statistical signalprocessing and include adaptive algorithms for radar detection and estimation.His current research in these areas are being sponsored by the U.S. Air Force andARPA. In 1994 he was an ASEE/Navy Summer Faculty Research Fellow with theNaval Air Warfare Center, Aircraft Division, Warminster, PA, where he worked onhigh resolution radar detection problems.Dr. Raghavan serves as a regular technical reviewer for these Transactions and

several other journals.

Haifeng Qiu was born in Wuhan, Hubei, People’s Republic of China, onSeptember 10, 1968. He received the B.S. degree in electrical engineering fromSouth China University of Technology, Guangzhou, People’s Republic of China, in1990.Since 1992 he has been with the Department of Electrical and Computer

Engineering, Northeastern University, Boston, MA, where he is currentlya Research Assistant and is working towards the Ph.D. degree in electricalengineering. His research interests include detection and estimation, statisticalsignal processing, and multirate signal processing.

David J. McLaughlin (S’82–M’89) was born in Haverhill, MA on August30, 1962. He received the B.S. and Ph.D. degrees in electrical and computerengineering from the University of Massachusetts, Amherst, in 1984 and 1989,respectively.He was employed at the Microwave Remote Sensing Laboratory (MIRSL),

University of Massachusetts, from June 1983 until September 1989, when hejoined the Department of Electrical and Computer Engineering, at NortheasternUniversity, Boston, MA, where he is currently an Assistant Professor. Duringthe summers of 1990 and 1991, he was an A.S.E.E. summer faculty researchfellow at the Radar Division, Naval Research Laboratory, Washington, DC. Hisresearch interests are microwave remote sensing, electromagnetic scattering, radaroceanography, and advanced radar systems.He is a member of the IEEE Geoscience and Remote Sensing, Microwave

Theory and Techniques, Antennas and Propagation, Aerospace and ElectronicSystems, and Ocean Engineering Societies and is a member of the AmericanGeophysical Union. He reviews papers for the IEEE Transactions on Antennas andPropagation, the IEEE Transactions on Geoscience and Remote Sensing, the IEEETransactions on Education, and Radio Science. He is a member of Tau Beta Pi andEta Kappa Nu.


Date post:	02-May-2023
Category:	Documents
Upload:	yazduni
View:	0 times
Download:	0 times

CFAR Detection In Clutter With Unknown Correlation Properties

Documents