Performance analysis of two covariance matrix estimators in compound-Gaussian clutter

F.Gini and J.H.Michels

Abstract: The authors present a thorough performance analysis of two covariance matrix estimators, the sample covariance matrix estimator (SCME) and the normalised SCME (NSCME), which are employed by adaptive radar detectors in Gaussian and compound-Gaussian clutter. Theoretical performance predictions are derived compared with the modified Cram&-Rao lower bound and checked with real-life sea clutter data. The results of the analysis show that the NSCME has superior performance in compound-Gaussian clutter and its performance is insensitive to the clutter multivariate distribution within the range cell under test and to the shape of the clutter correlation among different range cells. Conversely, the performance of the SCME heavily depends on the clutter distribution and has a dramatic worsening in spiky non- Gaussian clutter.

1 Introduction

Adaptive detection is important for modem radar systems which are tasked to operate in nonhomogeneous and nonstationary environments. While adaptive radar detec- tion against Gaussian noise has been largely investigated in the past (see, for example [l-5]), the same detection problem against a background of correlated non-Gaussian clutter has been investigated only recently [6-141. With the support of experimental data, the clutter process has been modelled as a compound-Gaussian process-specifically, as the product of a real positive random process, called texture, times a complex Gaussian process, usually called speckle. The texture represents the local clutter power whose fluctuations are induced by the spatial and temporal variations in the radar back scattering, while the speckle represents the properties of the coherent radar sensor 1151. For observation time intervals of the order of the coherent processing interval of the radar system, the texture can be considered a random variable, i.e. it fluctuates among different range cells, hut is constant within each range cell. In this case, the clutter process can be approximated by a spherically invariant random process (SIRP) 116-191.

The problem of adaptive array detection in such a non- Gaussian scenario has been considered in [7], and later in [6, 9, 201. More recently, Richmond proved that Kelly’s generalised likelihood ratio test (GLRT) statistic is again obtained when the data matrix is modelled as a multivariate elliptically contoured (MEC) distribution [ 141, of which the case of compound-Gaussian clutter with completely correlated texture among all range cells is a special case.

0 IEE, 1999 IEE Pioceedings online no. 19990479 DO/: 10.1049/ip-rsn:l9990479 Paper first rcccived 1st October 1998 and in revised form 5th March 1999 F. Gini is with the Department of Information Engineering, University of Pisa, Via Diotisalvi, 2-56126, Pisa, Italy J.H. Michels is with the Air Force Research LahoratorylSensors Directo- ratelSNRT, 26 Electronics Parkway, Rome NY 13441-4514, USA

IEE Proc.-Radar: Sonar Novig.. VOl 146, No. 3, June 1999

Different adaptive detection algorithms have been proposed in Gaussian and compound-Gaussian clutter; most of them make use of secondary data from adjacent range cells tn estimate the clutter covariance matrix, but the estimation algorithms are different. In this paper we consider two of them, namely the sample covariance matrix estimator (SCME) employed, among others, by Kelly’s GLRT [I] and Rohey’s adaptive matched filter (AMF) [41, and the normalised SCME (NSCME) employed by the adaptive linear-quadratic detector (ALQD) proposed in [9]. Comparative performance analy- sis of Kelly’s GLRT and the ALQD in terms of probability of false alarm and probability of detection has already been carried out in [9], but a separate performance analysis of the two covariance matrix estimators has not yet been reported.

The goal of this paper is to analyse in detail the behaviour of the two covariance estimators.

2 Problem motivation

In this section, we provide a brief description of the compound-Gaussian model as well as the covariance esti- mators. Let us assemble the rn complex samples of the signal received from the range cell under test (CUT), { z [n ] , n = 1,. . . , rn}, into the m-dimensional vector z= z/+jzQ= [z[l]. . .z[m]]: often called the primary data vector, where z, and zQ represent the vectors of the in- phase (4 and quadrature (Q) components. The interpulse interval is the reciprocal of the pulse repetition frequency (PRF). The detection procedure is given by the decision between the two hypotheses Hu and H I after the vector z has been received. Under the iiull hypothesis, Hu, it is assumed that the primary data consist only of clutter, i.e. z = c. According to the compound-Gaussian model, each element of c can be interpreted as the product of two independent random variables (rv), such that [ 16-18]

e = fix (1)

with x=x i+ jxQ - CN (0,ZM) being an m x 1 complex Gaussian circular random vector, with normalised covar-

133

iance matrix M (i.e. [MIjj = I) . x is the speckle, and 7 is the texture, which represents the local clutter power in the CUT. x, and xp are independent and identically distributed (IID), zero mean, Gaussian vectors with covariance matrix M . Given a specific value of I , e is a complex Gaussian circular vector. Thus, under the null hypothesis, the condi- tional covariance matrix of z is given by

E(zzHIr ,HO] = E(cc"l7) = ZE(c,cTIr) = 2 E ( c ~ c ~ l r }

= 2rM (2)

where E{ .) denotes statistical expectation. The uncondi- tional covariance matrix is Mz=E{zzHIHo) = 2 pM, with p A E { 7 } , Under the alternative hypothesis, H I , the data consist of the sum of signal s = n p and clutter c. c( is a complex unknown parameter, whilep is a known complex vector (the so-called steering vector).

When the clutter is Gaussian distributed, the adaptive array radar detection problem has typically two unknowns, a and M,; in compound-Gaussian clutter the problem presents additional unknowns: the parameters of the texture probability density function (pdf) pT(?) . To compensate for the ignorance of the nuisance parameters it is usually assumed that K blocks of signal-free secondary data, {zh= f i x * } ; = I, from K adjacent range cells are available. Secondary data are assumed to have the same pdf and correlation properties of the clutter in the CUT [l , 2 , 4 ] . Moreover, the speckle vectors {xh} are assumed IID, while the texture samples %an be partially correlated with correlation sequence R,[d = E { I ~ ih+, } , so {zh} are ortho- gonal but not independent, save for the case of IID {T~}, that implies R,[/] =E{&}S[ l] , where S[.] is the Kronecker delta symbol. Remark 1 : None of the above cited papers have considered the possible presence of a strong target signal in one or more of the reference cells, which would introduce a bias in the covariance estimate. This problem has been inves- tigated by many researchers under the assumption of IID samples with Rayleigh distributed envelope (see [21], pp. 255-263, and references therein for further details), hut not under the modelling assumptions reported in this paper. In this paper, we do not investigate further along this direc- tion. Rather, the estimation problem for a multitarget non- Gaussian scenario is left as the subject of futnre research.

The first covariance estimator we consider is the so- called sample covariance matrix estimator (SCME); it is the maximum likelihood (ML) estimator of M, in Gaussian clutter and is given by

A

( 3 )

This algorithm is used by many adaptive detectors [I-5, 141. For example, Kelly's approach to adaptive array detection in Gaussian clutter was to assume that (zk} are Gaussian IID and to use the GLRT approach; i.e. to replace thc unknown pararncters with their ML estimates. This approach results in a detection strategy that can he expressed as [ I ]

where 1 is a threshold coefficient set to give a prescribed probability of false alarm. The estimator of eqn. 3 is the ML estimator of M, in Gaussian clutter but not in compound-Gaussian clutter [I], save for the case of completely correlated { I ~ } , i.e. 7 = z I = T * = . . . = z h . In

134

fact, Richmond recently proved that, in this particular case, M Z is again the ML estimate of M, and the decision strategy of eqn. 4 is the GLRT detector. In fact, the completely correlated r case is equivalent to the Gaussian case for a given realisation of data across all range cells. Unfortunately, in modern radar systems operating at low grazing angles and with high resolution capabilities, the texture generally fluctuates (decorrelates) from one range cell to the other [15, 221, so that the above assumption is violated. As a consequence, Kelly's GLRT loses its constant false alarm rate (CFAR) behaviour and suffers a considerable detection loss [I)].

The other covariance estimator we consider here is a normalised version of the SCME; it is given by

where ik= (zfzh)l(2m) is the sample estimate of the clutter power in the k-th secondary range cell. Note that M, and M are equal to within the constant term 2.~1. In the following, we refer to estimator ( 5 ) as the normalised SCME (NSCME). It was derived in [7], and later used also in [6,9] as part of an adaptive detection algorithm designed to operate against componnd-Ganssian clutter. Specifically, in [9] the NSCME was used in the following decision strategy, called the adaptive linear-quadratic detector (ALQD):

where 1 is a threshold coefficient. The ALQD has the desirable property of being CFAR with respect to the clutter amplitude probability density function (apdt) and quite insensitive to possible mismatch between the design and actual clutter covariance structure. Remark 2. fi of e n 5 is statistically equivalent to

on pT(r ) nor on R,[/]. This is not true for the estimation algorithm of eqn. 3. Remark 3: In Kelly's GLRT, for large numbers of second- ary data (K + CO) M , approaches M, and the term in brackets goes to unity. Thus, the test reverts to the well known whitening matched filter, which is optimum in Gaussian clutter hut with large performance loss in compound-Gaussian clutter [17, 23, 241.

S= (mlQX;= I (xhxk)I(xfxk), 3 . so its pdf depends neither

3 Performance analysis

In this Section, we present a performance analysis of the two estimators in eqns. 3 and 5 against compound-Gaus- sian clutter. The results presented in this paper represent an extension of those previously reported in [IO, I l l . Consider first the SCME of eqn. 3 and assume that the speckle sequences {xh[ i ] } are wide-sense stationary for each k. This assumption is reasonable in practice based on local stationarity within a given range cell.

3.7 Covariance matrix estimation: SCME The ijth element of the sample covariance matrix is given hY

IEE Proc.Rodar: Sonor Navig.. Yo/. 146, No. 3, June 1999

It is also an unbiased estimator in compound-Gaussian clutter. In fact, recalling that zk[i] = & xi[i], we easily obtain

I K

K i = I ~ { f i ~ [ i , j ] ) = - C ~ [ z ~ [ i l z z b l ) = ~ ( z ~ x ~ [ i ] x ~ [ i l ]

= 2pRx[i - i] (8)

where p = E { z } and Rx[rl~-{x[nIr*[n+11}/2= E { x ~ [ n ] x ~ * [ n + 1 ] } = E { x ~ [ n ] x ~ * [ n + I ] } is the covariance sequence of the speckle Z and Q components. The mean square value of &[i, j] is given by

(9)

where we have used the assumption thatAvectors { x k } are IID with RA01 = 1, and defined p,[o = R,[l]/R,[O], the texture normalised autocorrelation sequence. Because the estimator in eqn. 3 is unbiased, its mean square error coincides with the variance and is given by

mse(k,[i,jl) = v a r [ ~ ~ i , j ] j = Ellkz[i,jI12) ~ I E ~ ~ J ~ , ~ I ) I 2

4p21Rx[i - ill2 (10)

Note that eqn. IO does not depend on the number m of samples collected from each range cell. A case study: rk=AR(l ) Gamma distributed random sequence. When the texture is Gamma distributed the clutter is K-distributed [lo, 15-17, 201. The texture pdf is then given by

process, as in 1251, i.e. the normalised covariance sequence has exponential shape:

(12) where pz is the one-lag correlation coefficient. A general- isation that extends the traditional SIRP model to account for partially correlated texture is reported in 1261. Inserting eqn. 12 in eqn. IO, we obtain

4 2 mse(A,[i,j]) = - Kv

(13)

The two extreme cases of completely uncorrelated and completely correlated texture (along the range direction) are obtained by setting pz=O and p r = 1, respectively. Note that the estimator of eqn. 3 is consistent in the mean square sense (mss), because it is unbiased and its variance goes to zero when K + w , save for the case p7= I . In fact, from eqn. 13 we have

and K+mp,+l lim lim mse(&,[i,j]} = (4p2/v)lR& ~ i l l2 # O . This is

a manifestation of non-ergodicity when averaging over range cells; in fact, for pT = 1, i is a constant for all range cells on any given realisation (this is the case considered in [14]). However, i varies randomly from realisation to realisation. Thus, averaging over range cells will not he equivalent to averaging down the ensemble.

When pz = 1, the estimator of eqn. 3 is mss consistent only when the clutter is Gaussian distributed, i.e. when v --f 00. For Gaussian clutter, from eqn. 13 we obtain mse(fi,[i,j]} =4y2/K, which does not depend on RX[& Numerical results: For ease o f comparison we derived all the numerical results by setting p = 1/2, so that Mz = M . Without loss of generality, consider the two elements m,[l,l] and m,[1,2]. In Fig. 1 we investigate the behaviour of the mse of &[I ,I] as a function of v for different values

O p..,,

where r(.) is the Gamma function and p is the mean value; that is, pc=E{i}, and v is the order parameter which is a measure of the deviation from Gaussianity. Alternatively, l l v is a measure of the clutter spikiness. Experimentally observed values of v span the range [O.1,+ w]. When v + w , the clutter becomes Gaussian distributed 115- 171. A simple expression for the mse derives when the texture is modelled as a first-order autoregressive (AR(1))

IEE hae.-Rodai: Sonor Navig, Vol 146, No. 3, June 1999 135

0.1 ... ... ..... ....... -._ ..... ........................

0.01 12 24 36 48 60 72 64 96 108 120

K

Fig. 2 SCME: mse(fiz[l,l]) against Kfor pl=O " YaIucS:

~ 0.5 I I0 50

_ _ _ _ ~ ~~~

. . . . .... fm

..................

-.-.- ....... ......... t ......... 0.01 12 24 36 48 60 72 84 96 108 120

K

-.-.- ....... ......... t ......... 0.01 12 24 36 48 60 72 84 96 108 120

K

Fig. 3 SCME: mse(fiz[l,l]) against Kfor pr= 1 "values: __ 0.5

1 I0 50

~~~~-

~~~

. . . . .... +m

of pT. These curves show the dramatic increase in the mse as the clutter becomes spikier, i.e. as v decreases. Thus, performance prediction based on the Gaussian assumption can be largely optimistic, especially when the texture is completely correlated from one range cell to the other (pz= 1). We also observe that the mse increases with pz save for the case o f Gaussian clutter (v t 00). For clutter with high spikes (low v) the mse increases with p7, Whereas, when v + 00, the mse is independent of p7. Note that mse(i,[l,l]) does not depend on the shape of R&], while it heavily depends on the shape of p,[&

The effect of the secondary data size K is investigated in Figs. 2 and 3. It is interesting to observe that when p z = 0 (Fig. 2) the m e decreases to zero for increasing values of K, while for pz = 1 (Fig. 3) it reaches the asymptotic value 4b2h = llv that is finite nonzero save for the case of Gaussian clutter (v + 00). The mse of h,[1,2] presents similar trends, with the only difference that it also ppends on the speckle one-lag correlation coefficient px= RAl]l

The behaviour of m~e(l;2~[1.1]) and mse(iZ[l,2]) as a function of px was also investigated. The results are reported in Figs. 4-7 for the two extreme cases p r = O

RAOI.

136

0.1

0.01 0 0.2 0.4 0.6 0.8 1 .o

PX

Fig. 4 NSCME and SCME: mse(fi[l,l]) and mse(fi,[l,l]) against pr for

SCME, Y = 10 SCME, Y = +m MCRLB

0.1

0 0.2 0.4 0.6 0.8 1 .o PX

Fig. 5 NSCMEandSCME: mse(fi[l,ll) ondmse(&,[l,l]) againsip& m= 16, K = 3 2 , p7= 1 ~ NSCME _ _ _ _ ~ SCME, u=O.5 --A-- SCME,v=l _ - ~ SCME, " = I 0

SCME, Y = +m MCRLB

I E

. * . * " * 0 0.2 0.4 0.6 0.8 i .n

PX

Fig. 6 NSCME and SCME: mse(&Il,21) and mse(fiz[l,21) against p&r m = 16, K=32, pI = 0 ~ NSCME ~ _ _ _ _ SCME, Y = 0.5 --A-- SCME, u = l

SCME, Y = 10 SCME, v = +m + MCRLB

/EE Pme.-Rador, Sonor Nnvig, Yol. 146, No. 3, June 1999

t

\, 0 0.2 0.4 0.6 0.8 1 .o

PX

Fig. 7 NSCME and SCME: me(fi[l,Z]) und mse(fi,[l,ZI) againsf p,/or m = 16, K=32, pi= 1 ~ NSCME

--A-- SCME, Y = 1 ~~~ SCME, Y = I0

+ MCRLB

~~~~~ SCME, = 0.5

SCME, Y = +M

(Figs. 4 and 6) and p7 = I (Figs. 5 and 7). It is worth observing that mse(mz[l,l]) does not depend on px and is less sensitive to changes of v when p7 = 0 rather than when pz = 1. On the other hand, mse(fi,[ 1,2]) depends on px, hut it is much more sensitive to changes of px when pr= 1 rather than when p 7 = 0 . Moreover, for a given px, this sensitivity decreases when the clutter becomes less spiky (increasing values of v) and it does not depend on px at all when the clutter is Gaussian distributed. In Figs. 4-7, the modified Cramtr-Ran lower hound (MCRLB) is also reported as a benchmark for any unbiased estimator' of the clutter covariance matrix. It represents a useful lower bound in those cases where the tme CRLB cannot he derived in closed form (271. The calculation of the MCRLB is outlined in the Appendix. Figs. 4-7 show that mse(hz [1,1]) reaches the MCRLB only when v + CO, whatever the value of px, while mse(fi, [1,2]) reaches the bound only when v + CO and px + 1 .

3.2 Covariance matrix estimation: NSCME Consider now the NSCME of eqn. 5. The estimate of the ijth element is given by

d where fk=(l i2m) E?=, Izk [ill' and = means that the right-hand term and the left-hand term have the same distribution (the same pdf). This equivalence can be obtained by replacing zk[i] with ,& xk[i] in h[i,j] and observing that 7 k cancels out. As a result the pdf of h[i,j] does not depend on pz(7) and R,[& Moreover, fi[i,j] is mss consistent for ( m , Q + (CO, CO> In fact, ik is mss consistent (i.e. lim E{(?, - 7 k ) } = O), so when m + CO we have h[i,j] + (112K) zk [il zk* L i l l ~ k = (1120 $= xk [i] xk* U] i.e. an mss consistent estimator of m[i,j] = R A - i] for K increasing to infinity. Note that f o r i = j a n d m = l , f r o m e q n . 15 weobtainfi[i,i]=l.This result implies a perfect estimate, and thus m = 1 represents

IEE Pmc.-Rada,: Sonar Navig., Yo/. 146, No. 3, June I Y Y Y

o>+m

a case of no practical interest. For higher m, the mse first tends to increase and then decreases to zero. Unfortunately, closed-fonn expressions for the mean square error of fi[i,j] with finite sample size are not available. Thus, we resorted to simulation. Numerical results: Results obtained by averaging over 10' Monte Carlo runs are plotted in Figs. 4-9. It was observed that the bias of eqn. 15 is quite insensitive to the value of K, while it depends on m and on the shape of Rx[& Conversely, the variance of &[iJ decreases monotonically with K. Since estimator eqn. 15 is biased, it is more useful to show curves of the mean square error, expressed as mse(fi[i,j]) = lbias(fi[i,j])lz+ var{fi[i,j]}.

Performance comparisons between the two estimation strategies are reported in Figs. 4-7. The mse ofNSCME is always lower than that of SCME, but in the case of Gaussian clutter they are very close. The plots in Figs. 4-7 also show that mse (h[ l , l ] ) and mse (fi[1,2]) are lower than the MCRLB for some values of p.. This anomalous situation is due to the biasedness of estimator eqn. 5. The fact that the mse of a biased estimator can be lower than the CRLB for unbiased estimators is also pointed out in [28], p. 73. Figs. 8 and 9 show curves of mse as a function of m for different values of the ratio Kim.

0.1 c

4 6 8 IO 12 14 16 18 20 22 24 rn

I

0.0011 " " " ' " " ~ " ' " " " " ' ' " ' ' " ' " ' " ' 4 6 8 10 12 14 16 18 20 22 24

rn

Fig. 9 NSCME: mse(?k[l,21) againsl m . b r pX=O.5 Klrn vducs: - 2

3 4 5

- - * - - 6

_ _ _ _ _ . . . . . . . .

137

For a given m, the mse decreases with Klm, i.e. with K, while for a given value of K/m the mse decreases with m. It was also observed that the decreasing rate is higher with smaller values o f p , and that mse(&[l,l]) and mse(&[l,2]) are almost insensitive to changes of px, save for large values of it (see Figs. 4-7). px= 1 represents a degenerate case for which xa[i] =x&] = xa, V i,j and k, so we have &[i,j] = (m/K) Xik"= I (Ixk12/XT= I lznl2) = 1 , i.e. the estimate is not affected by any error.

4 Results with lPlX sea clutter data

We checked our performance prediction with real sea clutter data collected at Osbome Head Gunnery Range (OHGR) in November 1993, with McMaster University IPIX radar [29]. IPIX is an experimental instrumentation class radar, capable of dual polarised and frequency agile operation. The data of the OHGR data sets are stored as I byte integers, from 0 to 255. All data records contain like polarisation (HH and VV) and cross polarisation (HV and VH) coherent reception. The radar PRF is 2 KHz and the pulsewidth is 200 ns; thus the range resolution is 30 m and the range acquisition window is 210 m wide. There are seven range cells, and the number of time samples per cell is N = 131072. The heamwidth is 0.9".

Statistical analysis of these data is reported in [22]. For the data set we analysed, the 16 frequencies in the X-band (9.4 GHz) are transmitted with the frequency agility mode (i.e. the frequency is changed on a pulse-to-pulse basis); the azimuth is fixed at 79.753" and the grazing angle is 0.945". The analysed data set was stored on November 12 with a sea state 3 (Beaufort scale), a wind speed of 22 km/h (moderate breeze condition) from a direction of 40" with respect to the looking radar direction and a significant wave height of 1.42 m. The average wave period was about 5 s. The analysis of amplitude histograms, of cumulants and moments has demonstrated that for VV polarisation the data distribution approaches the K model [22], whereas the cross polarisation reveals the presence of additive Gaussian thermal noise, and the HH polarisation data are better fitted by the log-normal distribution.

To test the performance of the two covariance matrix estimators we used the K-distributed VV polarised data, with shape parameter v = 1.25, scale parameter p = 7.5 x lo-' and speckle one-lag correlation coefficient px=0.93 (estimated on the entire set of data [22]). Since we have seven range cells an4 as a rule of thumb, it should be at least K = 2m, we set m = 2 and m = 4, and we estimated bias and variance of &,[l,l], &Jl,2], &[l , l ] and &[1,2], averaging over N/2=65536 and N/4=32728 estimates, respectively.

Tables 1 and 2 show the results obtained by processing the real sea clutter data with the estimator eqns. 3 and 5, respectively, and the corresponding values obtained by the analytical expressions (for the SCME) and by Monte Carlo simulation (for the NSCME). The results show good agreement between performance prediction based on the K model and the performance obtained by processing the real data. In particular, the results in Table 1 confirm that the algorithm in eqn. 3 is unbiased; in fact, << var. Moreover, the variance always lies between the two extreme cases of pr = 0 and pz = 1; this is also an indicator that the texture is partially correlated among different range cells. The agreement between the results obtained by processing real data and synthetic data is a good indicator of the validity of the performance prediction results reported in this paper for the other values of the clutter parameters.

5 Concluding remarks

In this work, we have considered the problem of estimating the clutter covariance matrix for adaptive radar detection. The contribution of this paper is to present a thorough performance analysis of two covariance matrix estimators previously proposed in the literature: the sample covar- iance matrix estimator (SCME) and the normalised SCME (NSCME). The performance of the two estimation strate- gies has also been compared to the modified CRLB, which represents a lower bound for the mean square error of any unbiased estimator. Performance analysis presented in this paper is based on closed-form analytical expressions, on

Table 1: SCME: bias and variance of h, H.ll and h7 11.21

Real data Analytical results

Bias Variance P z = O & = I

~~~

m=2 mz[l , l l 6 . 1 6 ~ io-' 0.5012 0.3714 1.0571

m, 11 21 1 . 6 0 ~ 1 0 - ~ 0.4942 0.3569 0.9956

m=4 mz[ l , l l - 1.06 10-4 0.5019 0.3714 1.0571

m> 11 2 1 - 1.76 x 0.4940 0.3569 0.9956

Table 2: NSCME: bias and variance of m[l,l] and m[1,2] ~~ ~

Bias Variance Real data Simulation Real data Simulation

m=2 m[i,f] 4 .13x10-4 5 .94x10-4 9 .57x10-3 1 . 6 9 ~ 1 0 ~ ~

h[1,2] - 5 . 3 8 ~ io-' - 1 . l O x IO- ' 1 . 6 6 ~ 2 . 8 9 ~ io-'

m=4 h[1,1] 4.77 x 10-2 3 .40x10- * 4 . 3 9 ~ 1 0 - ~ 4 . 4 3 ~ 1 0 - ~

h[1,2] -3.10 x I O - * -8.35 x io-' 2 . 8 6 ~ IO-' 3 . 9 4 ~ IO-'

138 IEE Proe~Rodrrr: Sonar Noviz.. Yoi. 146, No. 3. June 1999

simulated data and on real sea clutter data. The results shown here (and those not reported for lack of space) suggest the following conclusions: 1. The SCME is unbiased, and its msc does not depend on the number m of integrated samples and on the shape of the speckle covariance sequence. The mse increases with the texture one- lag correlation coefficient p T , and the mse of &,[l,2] also increases with p x , save for the Gaussian cluner case. The mse decreases with the clutter order parameter 1' (i.e. when the clutter becomes less spiky) and with the number K of secondary range cells. When the clutter texture is highly correlated among different range cells (i.e. pT E l), the mse cannot be reduced beyond a given value by increasing K, save for the case of Gaussian clutter (v + 00) . The mse is always greater than the MCRLB and the mse for the diagonal elements approaches this bound when v i oci. 2. The NSCME is biased. The bias is quite insensitive to K but it approaches zero when m goes to infinity (the estimator is asymptotically unbiased). The bias depends on px and is maximum for px".0.9. The mse is quite insensitive to px, save for px> 0.7 and reaches its maxi- mum at px" 0.9. The error variance decreases with m and with the ratio Klm, while the mse decreases with Kim. For any given m and K, the NSCME outperforms the SCME, for all values of v and p r , save for the estimation of the diagonal elements when Y + 00 and px is close to 0.9. The mse is very close to the MCRLU.

6 Acknowledgments

The authors wish to thank Dr. Murali Rangaswamy and Prof. Alfonso Farina for reading the manuscript and providing many useful comments. Thanks also to Dr. Anastasios Drosopoulos who kindly provided the database and the reference [29].

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RANGASWAMY, M., WEINER, D.D., and MICHELS, J.H.: 'Innova- tions based detection algorithm for correlated non-Gaussian random processes'. Proc. of 6th SSAP Workshop on Statistical Signal and Array Processing, 1992, Victona,, British Columbia, Canada, pp. 7-9 RICHMOND, C.D.: 'A note on non-Gaussian adaptive array detection and signal parameter estimation', IEEE S i p d Process. Lett., 1996, 3, (8). pp. 251-252 WARD, K.D., UAKER, C.J., and WATTS, S.: 'Maritime riirveillance radar. Part 1: Radar scatterine from the ocean surface'. IEE Proc. E

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Appendix: Calculation of MCRLB

It is well known that the Cramkr-Ran lower bound (CRLB) provides a benchmark by which the performance of each estimator is judged. The CRLB is obtained by inverting the Fisher information matrix (FIM), [28] Section 3.3, whose ijth element is given by

where 0 is the vector of the unknown parameters andp,(z;0) is the joint pdf of z, the data vector. E={.} denotes statistical expectation with respect top,(z;0). In our case, 0 T contains T T T - the elements of the clutter Covariance matrix, z = [zi 22 . . .ZK] IS

the Km-dimensional vector obtained by appending the Km- dimensional vectors zk, andp,(z;O) is given by

(17) Aerosp. Elechon. Syst, July 1999, 35, (3)

Gaussian spherically invariant random processes'. Technical Report RL-TR-96-4, January 1996 Rome Laboratoiy Rome, NY

11 MICHELS, 1.11.: 'Covariance matrix estimator performance in non- Gaussian duner proc~sses'. Proc. of 1997 National Radar Conference, May 1997, Syracuse, NY, USA, pp. 309-313

12 RANGASWAMY, M., and MICHELS, J.H.: 'A parametric multichannel detection algorithm for correlated non-Gaussian random processes'. Proc. of I997 National Radar Conference, May 1997, Syracusc, NY, USA, pp. 349-354

10 MICHELS, J.H.: 'Covariance inatrir estimator performance ill non- x p,(z,, z2,. . . , zK)drld.i, . . . dr,

where T = [rI T ~ . . . rK]' is the random vector whose elements ure the teXtUrC values in the K range cells. many practical situations the computation of the CRLB

out to be infeasible. Unfortunately, this is the case for the estimation problem considered in this as can easily he understood by inserting eqn. 17 in eqn. 16. We

IEE Pruc.~Roduu,: Sonar Navig., El. 146, *io. 3, June I Y Y Y 139

are not aware of a closed form expression of the CRLB for the covariance matrix estimation in compound-Gaussian clutter. In this paper we compare the performance of the two estimators of Sections 3.1 and 3.2 with the vector nzodij7ed CRLB (MCRLB) [27]. The latter relies on the definition of a properly modified Fisher information matrix, I d B ) , which is the expectation with respect to the random nuisance parameters (in our case the texture vector T) of the conventional FIM as computed for fixed nuisance parameters, i.e. I, (e) = E, {IM (T; e)}, where

Then, the MCRLB is obtained by inverting the modified

The conditional modified FIM of (eqn. 18) is easily obtained by recalling that, conditioned to T, the data are composed h K IID complex, zero mean, circular Gaussian vectors (z;~~=,, i.e.,

FIM Iu(0).

(19) Exploiting a result for correlated Gaussian observations [28], Section 5.2, we have

that does not depend on T. Under the assumption that the covariance matrix is symmetric the number of unknown elements contained in M i s Ne = m(m + 1)/2. For example, when m = 2 we have e=[m[l , l ] m[1,2] m[2.2]]' and Ne = 3. Assume the true normalised covariance matrix and its inverse are given by

Inserting eqn. 21 in eqn. 20 we easily obtain

The diagonal elements are the bounds we were looking for. This result can be extended to higher values of m. When the speckle process is modelled as an AR(I) process, i.e. [MI,,] = 1 5 i, j 5 m, it results that the error variance of any unbiased estimator of the elements of M should satisfy the following relationship:

140 IEE Pmc.-Rodor. Sonar Navip. Yo/. 146, No. 3, June 1999