a glrt for multichannel radar detection in the presence of...

p

tter

ussianr esti-nce areres arestimatescon-

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602-01-ocessing

Digital Signal Processing 15 (2005) 437–454

www.elsevier.com/locate/ds

A GLRT for multichannel radar detectionin the presence of both compound Gaussian clu

and additive white Gaussian noise✩

Bin Liu, Biao Chen1, James H. Michels

Department of Electrical Engineering and Computer Science, 3-114 Center for Science and Technology,Syracuse University, Syracuse, NY 13244, USA

Available online 14 March 2005

Abstract

Motivated by multichannel radar detection applications in the presence of both white Ganoise and Gaussian clutter with unknown power, we develop maximum likelihood parametemates for the disturbance process. Both cases with known and unknown white noise variatreated. As the estimators do not admit closed-form solutions, numerical iterative procedudeveloped that are guaranteed to at least converge to the local maximum. The developed eallow us to construct a generalized likelihood ratio test (GLRT) for the detection of a signal withstant but unknown amplitude. This GLRT has important applications in multichannel radar detinvolving both white Gaussian noise and spherically invariant random process clutter and isto have better detection performance and CFAR property compared with existing statistics. 2005 Elsevier Inc. All rights reserved.

Keywords: Maximum likelihood estimate; Generalized likelihood ratio test; Spherically invariant randomprocesses; Multichannel radar detection

✩ This work was supported by the Air Force Research Laboratory under Cooperative Agreement F302-0525. This paper was presented in part at the Second IEEE Sensor Array and Multichannel Signal PrWorkshop (SAM ’02), Rosslyn, VA, August 2002.

E-mail addresses: [email protected] (B. Liu), [email protected] (B. Chen), [email protected](J.H. Michels).

1 Fax: +1 315 443 2583.

1051-2004/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2005.01.010

438 B. Liu et al. / Digital Signal Processing 15 (2005) 437–454

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1. Introduction

Multichannel radar detection considers the detection for the possible presence oget at a given steering direction in background clutter/noise. For air-borne high resoradars operating at low gazing angles, the compound Gaussian process has emerviable model to describe the backscattering process. For this clutter model, the clutttor c is expressed asc = √

sg, whereg is complex Gaussian with covariance matrix� ands is a real nonnegative scalar, unknown clutter component (also called texture compstatistically independent ofg. The power variation of ground clutter among range cellcaptured by the variation ofs while the Gaussianity is dictated by the central limit theorapplied locally to each range cell. Thus, multichannel radar detection in the presecompound Gaussian clutter and additive white Gaussian noise (AWGN) can be formas the following hypothesis testing problem:

H0, x = √sg + n,

H1, x = av + √sg + n,

(1)

wherex ∈ CN is the complex observation data vector,N is the vector size,2 v is the steeringvector,a is the unknown complex signal amplitude, andn is the complex AWGN vectowith covariance matrixσ 2I. We further denoteM = s� + σ 2I as the covariance matrifor the overall disturbance for the cell under test. Here we considers as a deterministicunknown clutter parameter. However, it is worth mentioning that a widely referencedof random processes, the so-called spherically invariant random process (SIRP) is aclass of compound-Gaussian by imposing a stochastic parametric model on the scalExamples for SIRP clutter [1–5] include theK and Weibull envelope distributions fospecific shape parameter values.

While much effort has been undertaken in finding a good detector for signals embin compound Gaussian clutter, most existing work assumes a clutter-only model; i.presence of AWGN at the receiver is largely ignored. Consider the clairvoyant caknown �, i.e., the covariance structure ofg is known. In the absence of white Gaussnoisen, the maximum likelihood (ML) estimate of the unknown parameters, namelsignal amplitudea and the scalar power term for the compound Gaussian compons,can be readily derived. Substituting the ML estimate under the two hypotheses inlikelihood ratio for the hypothesis testing problem [4], one arrives at the well-knownstatistic in the form of

Γ1 = |xH �−1v|2(xH �−1x)(vH �−1v)

. (2)

We remark here that this test statistic has been independently developed in Ref. [5asymptotically optimum test for radar detection in compound Gaussian clutter usinrepresentation theorem for SIRP derived in Ref. [3]. Since this test statistic addedmatched filter detector a normalizing constantxH �−1x, we will term it the normalizedmatched filter (NMF).

2 In the context of space time processing,N = JL whereJ is the number of antenna elements andL is thenumber of pulses within one coherent processing interval.

B. Liu et al. / Digital Signal Processing 15 (2005) 437–454 439

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Disregarding the presence of the AWGN in the detection problem was largely basthe premise that clutter power is usually several orders of magnitude higher than thaAWGN, thus making the presence of AWGN seemingly irrelevant. It has been obsehowever, that the presence of AWGN causes the NMF statistic to lose the desiredproperty (see, e.g., [6,7]). Here we further demonstrate that, even under extremedisparity between clutter and noise, a carefully constructed detection statistic can scantly outperform that of (2) which neglects the presence of AWGN. To do so, we dein this paper a maximum likelihood parameter estimation procedure for the compGaussian model in the presence of AWGN. The developed estimates are used to coa new GLRT that takes into account both compound Gaussian and the white Gaussturbances. Throughout this work, we assume that the covariance matrix� is known as inRef. [4]. The problem of target detection in the mixture of compound Gaussian clutteAWGN has also been addressed by Gini et al. [8], wheres was considered as a randovariable and heuristic estimators of the texture component using moment matchindeveloped instead of the ML estimator as derived in this paper.

An invariance framework is considered in Refs. [9,10] for the signal detection proin Gaussian noise with covariance matrix of known structure but unknown level. Thcommon unknown scaling on the clutter and additive white noise was considered. Atension of this work to the problem in which unknown and distinct levels are applithe clutter covariance and white noise individually was presented in Ref. [11] andrecently in Ref. [12].3 In Ref. [12], the test statistic of (2) was implemented with a ranr

projection matrix and performance was assessed in terms of the rank of the clutterance matrix. The GLRT approach addressed here is full rank.

An alternative to the GLRT approach is to inherit the test form (2) by substituting� withthe estimated total covariance matrixM. While this is a meaningful choice, it is largely sperseded by the GLRT approach. The merit of the GLRT is precisely due to its applicawhen the texture component is unknown, thus the extra data dependent normalizinstant provides the necessary robustness and in particular the CFAR property for thWhens is known and henceM can be constructed, one should instead use the optilikelihood ratio test. This is precisely the GLRT approach that we adopt wheres is esti-mated and then used in the LR statistic. We note that the original NMF test (2) can aderived using the GLRT approach when the AWGN is not considered [4].

The GLRT approach assumes that for each test cell, the clutter texture componeunknown constant. As such, a deterministic ML estimate fors is developed. Alternativelyif an a priori distribution ons is available, a Bayesian approach can be adopted whstochastic ML estimate can be used to construct the GLRT statistic [13]. The drawbthe latter approach, in addition to its increased complexity, is the sensitivity to the ainformation—if the assumed prior distribution differs from that which truly governsdata, its performance may be severely degraded and becomes inferior to those statisdo not utilize such information. On the other hand, the developed GLRT can be appany compound Gaussian clutter model regardless of the prior ons. While this GLRT does

3 Reference [12] was published subsequent to the original submission of this paper.


vides

nown-formates.

es theng theeglects

r tex-itive

ere-

is de-ators.

chieveswnthe CR

,

-

a

,

not provide a uniformly most powerful (UMP) test, its performance nevertheless prosignificant robustness in the problem addressed here.

This paper is organized as follows. In Section 2, we derive the ML estimates of unkparameters for the model described in (1). As the estimators do not admit closedsolutions, we develop in Section 3 numerical procedures for finding the ML estimWe show that for large clutter-to-noise power ratio, the iterative algorithm guaranteconvergence to at least a local optimal point. In Section 4, a GLRT is constructed usideveloped estimators whose performance compares favorably with the NMF that nthe presence of white Gaussian noise. We conclude in Section 5.

2. ML estimate of compound Gaussian parameters

For airborne radar applications with compound Gaussian clutter, while the clutteture power (thes parameter) at the test cell is usually unknown, knowledge of the addwhite Gaussian noise power,σ 2, may be available from the operational system. We thfore distinguish the following two cases: (1)σ 2 known and (2)σ 2 unknown, and derivethe maximum likelihood estimation procedures. The Cramer–Rao (CR) lower boundtermined in Appendix A for these cases assuming the existence of unbiased estimSimulation results presented in Section 4 suggest that an unbiased estimator that athe CR bound may exist for the knownσ 2 case. However, the results for the unknoσ 2 case indicate that the estimation procedure is biased. Thus, a comparison withbound is not meaningful for the latter case.

2.1. Known σ 2

For the known noise power case, we assume, without loss of generality, thatσ 2 = 1 asthe observations can be properly normalized. In this case,M = s� + I. We need, thereforeto solve the ML equations for (i)s under theH0 hypothesis; and (ii)s anda under theH1hypothesis.

2.1.1. ML estimate of s under H0UnderH0, the only unknown parameter involved iss and we have the likelihood func

tion:

L(s;x) ∝ 1

‖M‖ exp(−xH M−1x).

Since� is assumed to be positive definite and Hermitian,� can be diagonalized byunitary transformation (a.k.a., eigen decomposition)

� = U�UH ,

whereU is a unitary matrix and� is a diagonal matrix whose diagonal elements, sayλi

for i = 1, . . . ,N , are real positive. Then

M = s� + I = U(s� + I)UH .

From this, we get


ty

(6)

‖M‖ =N∏

i=1

(sλi + 1),

M−1 = U

(diag

(1

sλ1 + 1, . . . ,

1

sλN + 1

))UH .

Therefore

∂‖M‖∂s

=N∑

i=1

λi

∏j �=i

(sλj + 1) = Tr(�M−1)‖M‖, (3)

∂M−1

∂s= U

(diag

( −λ1

(sλ1 + 1)2, . . . ,

−λN

(sλN + 1)2

))UH = −�M−2, (4)

where Tr(A) is the trace of matrixA. The last equality follows from the unitary properof U. By taking the derivative ofL(s;x) with respect tos and setting it to 0, we get

∂L(s;x)

∂s= − 1

‖M‖2

∂‖M‖∂s

exp(−xH M−1x)

− 1

‖M‖ exp(−xH M−1x)xH ∂M−1

∂sx = 0.

From (3) and (4), we have

Tr(�M−1) = xH �M−2x. (5)

2.1.2. ML estimates of a and s under H1UnderH1, botha ands are unknown and the likelihood function is

L(a, s;x) ∝ 1

‖M‖ exp(−(x − av)H M−1(x − av)

).

Taking the derivative with respect toa and setting it equal to zero, we have

a = vH M−1xvH M−1v

. (6)

The ML equation regardings can be derived in a very similar fashion as that underH0hypothesis and we get

Tr(�M−1) = (x − av)H �M−2(x − av). (7)

Thus, the ML estimates fora and s are the solutions to the two nonlinear equationsand (7).

2.2. Unknown σ 2

If σ 2 is unknown, we also need to find its ML estimate underH0 and H1. The newestimates are developed below.


e

tes dotimate

2.2.1. ML estimates of s and σ 2 under H0UnderH0, we have the likelihood function:

L(s,σ 2;x) ∝ 1


Since

M = s� + σ 2I = U(diag(sλ1 + σ 2, . . . , sλN + σ 2)

)UH .

We get

‖M‖ =N∏

i=1

(sλi + σ 2),

M−1 = U

(diag

(1

sλ1 + σ 2, . . . ,

1

sλN + σ 2

))UH .

Therefore

∂‖M‖∂σ 2

=N∑

i=1

‖M‖sλi + σ 2

= Tr(M−1)‖M‖,

∂M−1

∂σ 2= U

(diag

(1

(sλ1 + σ 2)2, . . . ,

1

(sλN + σ 2)2

))UH = −M−2.

Taking the derivative ofL(s,σ 2;x) with respect tos andσ 2 and setting them to zero, wobtain

Tr(�M−1) = xH �M−2x,

Tr(M−1) = xH M−2x.

}(8)

2.2.2. ML estimates of a, s, and σ 2 under H1The likelihood function becomes

L(a, s, σ 2;x) ∝ 1

‖M‖ exp(−(x − av)H M−1(x − av)

)and by taking the derivative ofL(a, s, σ 2;x) with respect tos, σ 2, anda and setting themto zero, we get

a = (vH M−1x)/(vH M−1v),

Tr(�M−1) = (x − av)H �M−2(x − av),

Tr(M−1) = (x − av)H M−2(x − av).

(9)

3. Numerical procedure for solving the ML equations

The set of nonlinear equations developed in Section 2 for solving the ML estimanot admit closed-form solutions even for the simplest possible case, namely the es


hileto solvetermsR) islways

plicity

algo-

the

dr-

mpli-

ch

r equa-to the

of s underH0 with σ 2 known (Eq. (5)). Thus, we resort to numerical procedures. Wmany standard numerical methods such as the Newton method [14] can be appliedthese equations, we found that a simple bisection algorithm works reasonably well inof performance and efficiency, especially when the clutter-to-noise-power ratio (CNlarge. Indeed, for large CNR, the solutions to the nonlinear equations are almost aunique, which makes the bisection algorithm an appealing candidate due to its simin terms of implementation.

We use the simple example of (5) to illustrate the implementation of the bisectionrithm. Rewrite (5) as

f (s) � Tr(�M−1) − xH �M−2x = 0, (10)

where underH0, x ∼ CN(0,M). We use the following bisection method to obtainsolution off (s) = 0.

1. Finds−0 < s+

0 such thatf (s−0 ) < 0< f (s+

0 ). Setk = 0.2. If |s+

k − s−k | < ε for a given toleranceε, thensfinal = (s+

k + s−k )/2.

3. Else,sk+1 = (s+k + s−

k )/2; and• if f (sk+1) = 0, thensfinal = sk+1;• else iff (sk+1) < 0, thens−

k+1 = sk+1 ands+k+1 = s+

k ;• else iff (sk+1) > 0, thens+

k+1 = sk+1 ands−k+1 = s−

k .4. k = k + 1. Go to step 2.

We point out that even forε values chosen as low asε = 10−9, the bisection methousually ends in less than ten iterations. Thus,ε has little effect on the estimator perfomance.

The bisection algorithms for other sets of nonlinear equations are slightly more cocated. For example, with knownσ 2 and underH1, we need to estimatea in addition tos.However, sincea has a closed-form solution givens, one only needs to insert a step at eaiteration fora.

Next, we discuss the existence and uniqueness of the solutions to the nonlineations. We show that under the large CNR assumption, there always exist solutionsset of nonlinear equations. Consider the case of hypothesisH0 with known noise vari-ance, for which we only have a single variable to deal with. Forf (s) as in Eq. (10),we know that� can be diagonalized by a unitary matrixU, i.e., � = U�UH , where� = diag(λ1, λ2, . . . , λN). Thus, we have

f (s) = Tr(�M−1) − xH �M−2x

= Tr(U�UH (U�UH + σ 2I)−1) − Tr

(xH U�UH (U�UH + σ 2I)−2x

)=

N∑i=1

λi

sλi + σ 2−

N∑i=1

|yi |2λi

(sλi + σ 2)2

=N∑ λi

sλi + σ 2

(1− |yi |2

sλi + σ 2

), (11)

i=1


y)

rms in

roughined.

Fig. 1.f (s) as a function ofs. The trues equals 0.4565.

where we definey = UH x hence the covariance matrix fory is s0� + σ 2I with s0 beingthe true underlying scale parameter. Denote

gi(s) = 1− |yi |2sλi + σ 2

and fi(s) = λi

sλi + σ 2gi(s).

Thengi(s) is monotonically increasing ins with

lims→∞gi(s) = 1 and lim

s→0gi(s) = 1− |yi |2

σ 2.

DenoteP0 as the probability that the limit at zero forgi(s) is less than zero, i.e.,

P0 = P

(1− |yi |2

σ 2< 0

)= P(|yi |2 > σ 2).

Sinceyi ∼ CN (0, s0λi + σ 2) and for large CNR (i.e.,s0λ σ 2), P0 ≈ 1. Therefore ass → 0, we havegi(s) < 0 with probability close to one. Consequently,fi(s) is negativefor small s but approaches zero from the positive whens → ∞. From Eq. (11), it is easto see that there exists at least one solution forf (s) = 0. In fact, a close inspection of (11reveals that if the CNR is large for those dominant components (largeλi ’s), existence ofsolutions is guaranteed with probability close to 1. To see this, notice that those te(11) with largeλi dominate whens → 0.

As to the uniqueness, while analytic proof has not been obtained, it is found ththorough numerical simulation that for large CNR, a unique solution is always determFigure 1 is a typical example forf (s) as a function ofs where the trues0 = 0.4565. Thelow CNR case will be considered in future research.


tion ofarof an

Fig. 2. Cramer–Rao lower bound and mean square error as a function of unknown parameters.

Fig. 3. Mean value of the estimation as a function of unknown parameters.

Figure 2 shows plots of the mean squared error and CR lower bound as a functhe unknown parameters for the known and unknownσ 2 cases. Figure 3 shows a similplot for the mean value. As noted previously, these results indicate the existence


known

GLRT

NMFn the

edern,

ll,alized

er has.

utteres-es thea sig-ation,ence,e per-

t to the

hich

unbiased estimator that achieves the CR bound for the knownσ 2 case. Although this wouldimply the existence of a strictly data dependent estimator (i.e., independent of the unparameter), a solution of such an estimator is currently not available.

4. Performance evaluation

4.1. Performance comparison with NMF

The ML estimates developed in the previous sections can be used to construct a4

for the detection problem specified in (1):

Γ2 = maxa,s,σ2 f (x|a, s, σ 2;H1)

maxs,σ2 f (x|s, σ 2;H0)

= maxa,s,σ21

‖M‖ exp(−(x − av)H M−1(x − av))

maxs,σ21

‖M‖ exp(−xH M−1x). (12)

In this section, we use numerical examples to compare the proposed GLRT with thedeveloped in Refs. [4,5], which itself is a GLRT assuming clutter-only disturbance. Ifirst example, we use two channels, four pulses, henceN = 8, and the average CNR=40 dB. The output signal to interference and noise power ratio (SINR), defined as

SINR= 10 log10 |a|2vH M−1v,

is fixed at 6 dB. As noted in Section 1,K envelope distributed clutter is often modellusing an SIRP model. Here the clutter assumes aK distribution with a shape parametα = 0.1. Specifically, the texture components is generated using a Gamma distributioand it varies over range cell as well as from trial to trial. However, at each range ces isan unknown constant parameter. The clutter ridge lies along the diagonal in the normDoppler-spatial frequency domain. The target signal is located at 0◦ azimuth and 0.15normalized Doppler frequency in the spatial-temporal (Doppler) domain and the cluttone lag temporal correlationut = 0.999 which specifies the covariance matrix structure

As a motivation, we first compare the performance of NMF in clutter-only and in clplus noise (with average CNR= 40 dB) disturbance. As evidenced in Fig. 4, the prence of AWGN, even with power disparate with the clutter power, severely degradperformance. This is because the clutter-only rejection of NMF is accompanied bynificant degradation on the signal-to-white-noise ratio. Quantification of such degradhowever, seems impossible as the NMF does not admit a linear detector form. Hthe signal and noise components in the NMF output are generally inseparable. Thformance degradation motivates the search for new statistics that are more robuspresence of AWGN.

4 Alternatively, given the estimated parameterss andσ , one can reconstruct matrix estimateM and replace�with M in the test statisticΓ1. While viable, this approach is largely superseded by the proposed GLRT wuses the estimates directly in the LR statistic.


tistics,

antpulse

(a)

(b)

Fig. 4. The ROC curves of the NMF under clutter-only and clutter-plus-noise disturbances. (a)N = 8; (b)N = 64.

Figure 5a gives the receiver operating characteristics (ROC) curves of the two stanamely the NMF and the proposed GLRT statistic. For the cases of both knownσ 2 andunknownσ 2, the proposed GLRT of (12) outperforms the NMF of (2) by a significmargin. In the second example (shown in Fig. 5b), we use a two channel thirty-two


con-es form

(a)

(b)

Fig. 5. Performance comparison between the two GLRT (Γ1 andΓ2) in the presence ofK distributed clutter andAWGN. (a)N = 8; (b) N = 64.

example (henceN = 64) which is otherwise identical to the previous case. The sameclusion holds for this higher dimensioned case. The only difference is that the curvGLRT in the cases of knownσ 2 and unknownσ 2 are almost identical. This results fro


reell.rm atMF,

perty,

withesent,

para-abilitydB.e pa-

alarmfplen

within

icular

Fig. 6. Probability of detection as a function of SINR.N = 8 andσ2 is assumed unknown.

the improved estimation performance forσ 2 (and hences as the nonlinear equations acoupled) in the higher dimension case due to the increased data size for each test c

Figure 6 gives the probability of detection versus the SINR for a fixed false ala10−3 with N = 8 andσ 2 unknown. It is noted that the proposed GLRT outperforms Nespecially in the low SINR region.

4.2. Discussion of the CFAR property

In the absence of white Gaussian noise, the NMF of (2) has the desired CFAR proi.e., the false alarm rate is independent of the clutter power terms. In the context ofKdistributed clutter, the CFAR with respect to power variation implies that it is CFARrespect to the shape parameter. This CFAR property, however, is lost if AWGN is pras evidenced in Fig. 7, where the probability of false alarm as a function of the shapemeter is obtained via simulation. The threshold is chosen assuming a nominal probof false alarm at 10−3 in the clutter-only case. The average CNR is again fixed at 40Clearly, the probability of false alarm changes significantly as a function of the shaprameter in the presence of additive noise, indicating the loss of CFAR for the NMF.

In Fig. 8, using the same setting as in the first example, the probability of falseof the proposed GLRT is given for a fixed threshold for the knownσ 2 case. Notice that ithe clutter texture terms is known perfectly, then the problem specified in (1) is a simGaussian noise problem with known covariance matrixM and the detection statistic i(12) reduces to the matched filter for Gaussian disturbance. Hence it is clearly CFARrespect tos. The fact that we have to estimates changes the CFAR property as shownFig. 8a, most noticeably in the region with very small shape parameter. In this part


para-ate

s, theNR

larger

n

signalclutter.blemThecluttersimplees arelopedat the

Fig. 7. False alarm rate of the NMF statistic (Γ1) as a function of the shape parameter of theK clutter. ThenominalPf = 0.001.

example, we notice that the proposed GLRT is still CFAR with respect to the shapemeter when it is greater than 0.1. The reason can be explained as follows. The ML estimof s is likely to be very accurate for large CNR. At very low shape parameter valuevariance of the clutter texture terms becomes large. Therefore, even if the average Cis kept at 40 dB, the likelihood of having smaller CNR increases. This results in aerror variance of the estimate fors which in turn affects the CFAR property.

This CFAR performance will improve as the dimensionN increases. Illustrated iFig. 8b is a case forN = 64 that shows a better CFAR property than that ofN = 8. This isdue to the improved estimation performance fors using largeN , as mentioned before.

5. Conclusions

In this paper, we consider the detection problem for the case of unknown, constantamplitude in the presence of both white Gaussian noise and compound GaussianAdopting a deterministic ML approach, the problem is equivalent to the detection proin the mixture of AWGN and Gaussian clutter with unknown clutter to noise ratio.ML estimates of parameters associated with the detection problem, including thetexture component, the target amplitude, and the white noise variance are derived. Abisection algorithm is devised for solving the ML equations. The developed estimatthen used to construct a GLRT test which can be shown to outperform the NMF devefor the clutter-only case in both detection performance and CFAR property, althoughexpense of increased computational complexity.


icon

(a)

(b)

Fig. 8. Probability of false alarm of the new GLRT (Γ2) as a function of the shape parameter of theK clutter.(a)N = 8; (b)N = 64.

The GLRT approach, which treats the texture components as an unknown deterministvariable, has the desired robust property—it does not rely on the a priori distributions.Its detection performance, however, is largely determined by the ratios/σ 2 for a given cell.


ibution

ch hap-nfurther

proachn thevolv-GLRT

ted inpara-

On the other hand, a stochastic ML approach can be adopted when the a priori distron s is available. In such case, the optimal detector is not sensitive to the ratios/σ 2 on aper cell basis but is less robust as its performance is greatly affected when a mismatpens. That is, the assumed a priori distribution ons may deviate from the true distributiothat governs the data. A careful comparison between the two approaches deservesattention and will be considered in future research.

Further, and most important, the more realistic case of unknown� would require alarge number of unknown parameters for the approach considered here. The apin Ref. [12] considered a low rank approximation to eliminate the dependence upounknown parameters while that in Refs. [6,7] utilized a model based approach ining multichannel autoregressive parameters containing a reduced parameter set. Aimplementation based on the NPAMF method reported in Refs. [6,7] was presenRef. [15]. Augmentation of such approaches with knowledge of the unknown scalemeters may provide a future research direction.

Appendix A. Derivation of the Cramer–Rao lower bound

For the case of unknowns andσ 2, the likelihood function underH0 is

L(s,σ 2;x) ∝ 1


Then the log-likelihood function is

l(s, σ 2;x) ∝ −xH M−1x − ln‖M‖whose first derivative with respect tos andσ 2 are

∂l

∂s= xH �M−2x − Tr(�M−1),

∂l

∂σ 2= xH M−2x − Tr(M−1)

and second derivative with respect tos andσ 2 are

∂2l

∂s2= −2xH �2M−3x + Tr(�2M−2),

∂2l

∂σ 4= −2xH M−3x + Tr(M−2),

∂2l

∂σ 2∂s= ∂2l

∂s∂σ 2= −2xH �M−3x + Tr(�M−2).

The regularity condition is satisfied since

E

[∂l

∂s

]= E

[Tr(�M−2xxH )

] − Tr(�M−1) = Tr(�M−1) − Tr(�M−1) = 0,

E

[∂l

2

]= E

[Tr(M−2xxH )

] − Tr(M−1) = Tr(M−1) − Tr(M−1) = 0,
∂σ


in: IEE

lly in-

Trans.

E

IEEE

aussian

d filterria, VA,

ound-Signal

6–2157.g, in:ve, CA,

d addi-rkshop,

where we useE[xxH ] = M , the Fisher information matrix can now be written as

J (s, σ 2) =( −E

[∂2l

∂s2

] −E[

∂2l

∂s∂σ 2

]−E

[∂2l

∂σ2∂s

] −E[

∂2l

∂σ4

])

,

where

−E

[∂2l

∂s2

]= E

[Tr(2�2M−3xxH )

] − Tr(�2M−2) = Tr(�2M−2),

−E

[∂2l

∂σ 4

]= E

[Tr(2M−3xxH )

] − Tr(M−2) = Tr(M−2),

−E

[∂2l

∂σ 2∂s

]= −E

[∂2l

∂s∂σ 2

]= E

[Tr(2�M−3xxH )

] − Tr(�M−2) = Tr(�M−2).

The Cramer–Rao lower bound follows as

var(s, σ 2) � J−1(s, σ 2) =(

Tr(�2M−2) Tr(�M−2)

Tr(�M−2) Tr(M−2)

)−1

.

Similarly for knownσ 2, it follows that

var(s) �[Tr(�2M−2)

]−1.

References

[1] E. Conte, M. Longo, Characterisation of radar clutter as a spherically invariant random process,Proc. F, Radar, Sonar, and Navigation, vol. 134, April 1987, pp. 191–197.

[2] M. Rangaswamy, D. Weiner, A. Ozturk, Non-Gaussian random vector identification using sphericavariant random processes, IEEE Trans. Aerospace Electron. Syst. 29 (1983) 111–123.

[3] K. Yao, A representation theorem and its applications to spherically invariant random processes, IEEEInform. Theory 19 (1973) 600–608.

[4] F. Gini, Sub-optimum coherent radar detection in a mixture ofK-distributed and Gaussian clutter, in: IEProc. F, Radar, Sonar and Navigation, February 1997, pp. 39–48.

[5] E. Conte, M. Lops, G. Ricci, Asymptotically optimum radar detection in compound-Gaussian clutter,Trans. Aerospace Electron. Syst. 80 (1995) 615–625.

[6] J.H. Michels, B. Himed, M. Rangaswamy, Performance of STAP tests in Gaussian and compound-Gclutter, Digital Signal Process. 10 (2000) 309–324.

[7] J.H. Michels, M. Rangaswamy, B. Himed, Evaluation of the normalized parametric adaptive matcheSTAP test in airborne radar clutter, in: Proceedings of the International Radar Conference, Alexand2000.

[8] F. Gini, A. Farina, M.V. Greco, Detection of multidimensional Gaussian random signals in compGaussian clutter plus thermal noise, in: Proceedings of the Fourth International Conference onProcessing, Beijing, China, November 1998, pp. 1650–1653.

[9] L.L. Scharf, B. Friedlander, Matched subspace detectors, IEEE Trans. Signal Process. 42 (1994) 214[10] L.L. Scharf, T.L. McWhorter, L.J. Griffiths, Adaptive coherence estimation for radar signal processin

Proceedings of the 30th Asilomar Conference on Signals, Systems, and Computers, Pacific GroNovember 1996.

[11] B. Liu, B. Chen, J.H. Michels, A GLRT for radar detection in the presence of non-Gaussian clutter antive white Gaussian noise, in: Proc. 2002 IEEE Sensor Array and Multichannel Signal Processing WoRosslyn, VA, August 2002.


eneous

es for), Radar

ample

[12] M. Rangaswamy, F.C. Lin, K.R. Gerlach, Robust adaptive signal processing methods for heterogradar clutter scenarios, Signal Process. 84 (9) (2004) 1653–1665.

[13] B. Ottersten, M. Viberg, P. Stoica, A. Nehorai, Exact and large sample maximum likelihood techniquparameter estimation and detection in array processing, in: S. Haykin, J. Litva, T.J. Shepherd (Eds.Signal Processing, Springer-Verlag, Berlin, 1993.

[14] D.P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, MA, 1995.[15] P. Li, H. Schuman, J.H. Michels, B. Himed, Space-time adaptive processing (STAP) with limited s

support, in: Proceedings of the IEEE Radar Conference, Philadelphia, PA, April 2004.

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