Time-averaged subspace methods for radar clutter texture...

1886 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 9, SEPTEMBER 2001

Time-Averaged Subspace Methods for Radar ClutterTexture Retrieval

Fulvio Gini, Senior Member, IEEE, Georgios B. Giannakis, Fellow, IEEE, Maria Greco, Member, IEEE, andG. Tong Zhou, Senior Member, IEEE

Abstract—Subspace approaches have become popular in thelast two decades for retrieving constant amplitude harmonicsobserved in white additive noise because they may exhibit superiorresolution over the FFT-based methods, especially with shortdata records and closely spaced harmonics. We demonstrate herethat MUSIC and ESPRIT methods can also be applied when theharmonics are corrupted by white or wideband multiplicativenoise. The application context is the retrieval of texture infor-mation from high resolution and low grazing angle radar clutterdata affected by wideband colored speckle that is modeled ascomplex multiplicative noise. Texture information is fundamentalfor clutter cancellation and constant false alarm rate (CFAR)radar detection. A thorough numerical analysis compares the twosubspace methods and validates the theoretical findings.

Index Terms—Clutter cancellation, frequency estimation,high-resolution radar, multiplicative noise, MUSIC and ESPRITmethods, radar clutter, texture retrieval.

I. INTRODUCTION AND PROBLEM STATEMENT

H ARMONIC retrieval in the presence of additive noise hasbeen a classical signal processing problem for about a

century. Recently, however, it has been realized that multiplica-tive noise may also appear in certain harmonic retrieval applica-tions. Multiplicative noise shows up in radar imaging due to thepresence of speckle noise [4], [20], [33]. It is also encounteredin weather radar and underwater acoustic applications, wherethe multiplicative noise could be due to the fluctuation of themedium [38]. Lightwave signals for digital communications,amplitude-modulated signals, and fading communication chan-nels also comply with the multiplicative noise model [10], [12],[13].

In this paper, the focus is on a radar application: the estima-tion of the sea clutter texture as sensed by a high-resolution radarsystem operating at low grazing angle. Under this circumstance,the observed signal can be modeled in discrete-time as

Manuscript received May 22, 2000; revised June 4, 2001. The work of F. Giniand M. Greco was supported by the Italian Space Agency (ASI) through ProjectZODA900167. The associate editor coordinating the review of this paper andapproving it for publication was Dr. Olivier Besson.

F. Gini and M. Greco are with Department of Information Engineering, Uni-versity of Pisa, Pisa, Italy (e-mail: [email protected]; [email protected]).

G. B. Giannakis is with Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]).

G. T. Zhou is with School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail:[email protected]).

Publisher Item Identifier S 1053-587X(01)07046-5.

(1)

with the following assumptions.

a1) s are deterministic positive constants.a2) s are deterministic constants in .a3) s are distinct and nonzero constants in .a4) Process is complex valued, stationary, and zero-

mean.The only constraint that is posed on , , and

is that they produce for every . Motiva-tion for the use of model (1) is detailed in Section II. Given

, we wish to retrieve , which amounts to esti-mating , and .

Since the mean of is nonzero, the cyclic mean of(1), i.e., the normalized Fourier transform of , can be usedto estimate the frequencies [9]. The limitation, however, isthat frequencies separated less than cannot be resolved[28, p. 264]. This motivates us to study subspace methods [28,Sec. 9.7], which have not been applied to the random amplitudemodel in (1). We prove in this paper that MUSIC and ESPRITmethods are capable of retrieving harmonics in the presence ofwhite or colored wideband multiplicative noise with unknownspectral density as well.

The organization of the paper is as follows. In Section II,we introduce the radar problem that motivates the use ofsignal model (1). In Section III, we briefly review the conceptof time-averaging statistics for mixed processes, and then,we use this concept to formulate MUSIC and ESPRIT forharmonics observed inwhite multiplicative noise. The useof subspace methods for retrieving harmonics in widebandcolored multiplicative noise is also discussed. Simulationexamples are provided in Section IV to illustrate the twoestimation algorithms and their ability to retrieve clutter textureinformation from intensity data. Finally, conclusions are drawnin Section V.

II. HIGH-RESOLUTIONRADAR SEA CLUTTER MODEL

The most popular statistical model in the radar literaturefor high-resolution and low grazing angle sea clutter is thecompound-Gaussianmodel [6], [30], [34], [35]. According tothis model, samples of the complex envelope of the sea clutterprocess are expressed as the product between two components:

. Factor isa complex Gaussian process, which is usually called the

1053–587X/01$10.00 © 2001 IEEE

GINI et al.: TIME-AVERAGED SUBSPACE METHODS FOR RADAR CLUTTER TEXTURE RETRIEVAL 1887

speckle, and accounts for local backscattering; andare the in-phase () and quadrature () components

that are assumed to satisfy with; thus, the speckle power is

. Factor is a non-negative realrandom process, which is usually calledtexture; it describesthe underlying mean power level of the data. Due to theirdifferent physical origin, these two components present dif-ferent correlation lengths. For X-band high-resolution data, thespeckle correlation length was measured to be on the orderof tens of milliseconds, while that of the texture was on theorder of seconds [8]. Although many hypotheses have beenformulated and tested on measured sea clutter data, they pertainto the one-dimensional (1-D) probability density function(PDF) of the texture and of the clutter amplitude [2], [5], [8],[19], [36]. In fact, because of its long correlation time, thetexture has been, so far, considered constant within each radarcoherent processing interval (CPI), changing according to agiven PDF from one CPI to the next, that is,for ; thus,is a spherically invariant random vector(SIRV) [31], [35].Conditioned on a given value of, the complex data vector

is Gaussian distributed. Thanks to this property, this modelis mathematically tractable and has gained popularity in theradar literature [11], [14], [15], [30], [34]. However, there is aprominent interest to model the texture as a process, in orderto predict its behavior in time intervals greater than a CPI,for clutter cancellation and constant false alarm rate (CFAR)detection purposes [3], [25], [26].

For ocean imaging, it is common practice to analyze and plotsamples of the intensity of the radar echo, that is, the modulosquared of the complex envelope samples, which are sometimesincoherently integrated over a small space-time window, tohighlight the local power and to reduce the speckle effect. If thesea is fully developed, the characteristic periodic wave structurewith alternating crests and troughs is clearly visible in therange-time or range-azimuth intensity maps, particularly in VVpolarization [26], [29]. For the compound-Gaussian sea radarclutter, the intensity is given by .Due to their different correlation lengths, the texture and thespeckle have been usually separated using lowpass filtering [8],[22]; in particular, the texture has been estimated (nonparamet-rically) by means of a moving-window filter of length as

(2)

In order for the approximation in (2) to be valid, parametershould be such that , where and are thespeckle and texture correlation lengths, respectively. The term

in (2) represents the sample estimate of the speckle powerwhose actual value is one. When the speckle is Gaussian dis-tributed, represents the maximum likelihood estimate of

given the data samples

Fig. 1. Clutter intensity and texture versus time;� = 0:1, SNR= �7:59

dB, SNR = SNR = SNR = �15:37 dB.

Fig. 2. Texture retrieved by means of the moving window (MW) estimator;� = 0:1 SNR= �7:59 dB, SNR = SNR = SNR = �15:37 dB.

. Note that in order to choose the value of, the speckleand texture correlation lengths and should be known, atleast approximately. A method for estimating them from mea-sured clutter data is reported in [8] and employed in [16].

We model here the non-negative sea clutter texture as a sumof cosinusoids and a constant term

(3)

so that (1) is obtained. Fig. 1 depicts a realization of andin the presence of complex Gaussian distributed speckle

with Gaussian-shaped power spectral density (PSD) and withparameters given inExample 1. In Fig. 2, we plot the estimateof , obtained by means of the moving window (MW) esti-mator of (2) for and ; it is evident that the estima-tion accuracy is not satisfactory.

Under a1)–a4), in (1) is (almost) cyclostationary1 [7],[9], [12]. The presence of cyclostationarity in the sea clutter has

1Cyclostationary processes are those whose statistics are periodic functionsof time. Almost, here, refers to the fact that these periods may not be integersas required by our discrete-time framework.


been already documented theoretically by Conte and Longo in[6] and empirically by Haykin and Thompson in [21]. Startingfrom the empirical evidence that high-resolution and lowgrazing angle sea clutter is cyclostationary, due to the periodicstructure of the sea surface [1], [27], [32], we model the texturecomponent of the clutter as the superposition of real cosineswith unknown frequencies, amplitudes, and phases. Model (3)has been also validated by processing measured high-resolutionsea clutter data in [17] and [18].

Remark 1: Estimator (2) implicitly assumes thaton a small time-window of size and attempts to estimatethe clutter power by making use of samples only(with ). This estimate is updated at each time instantbecause is locally constant, but the value of this constantchanges (slowly) with time during the observation interval oflength . It is worth stressing that the estimator in (2) does notmake any attempt to model the time-variation of the texture, i.e.,it is a nonparametric estimator. In (3), we model the texture timevariation by means of cosinusoids, and we expect that bymaking use of this additional information, we can improve onthe MW estimator of (2), and most importantly, we can predictits behavior outside the observation interval. Another advantageof the subspace methods to be developed in this paper is that theydo not require knowledge of the speckle and texture correlationlengths. In the next sections, we show how this improvementcan be obtained.

Remark 2: In (1) and subsequent analyses, we do not con-sider the contribution of additive thermal noise, mainly for tworeasons: i) In the radar application described in Section II, thethermal noise is negligible relative to the clutter power at leastfor polarizations VV and HH [8], [16]; ii) we are interested inthe multiplicative noise effect, which usually affects estimationaccuracy more than the additive noise (the multiplicative noise isresponsible for the well-known “floor” effect in estimation anddetection performance). Due to lack of space, we do not includehere numerical results for small clutter-to-thermal noise powerratio; however, there is no theoretical problem in including ad-ditive thermal noise to our analysis.

III. T IME-AVERAGED STATISTICS AND TIME-AVERAGED

SUBSPACEMETHODS

First, we stress that although different realizations of theclutter process may have independent and identically dis-tributed (i.i.d.) phases uniformly distributed in , inthe clutter time series analysis, only a single record is oftenavailable. Moreover, aiming at clutter cancellation and targetdetection, we want to retrieve only for the specific realiza-tion under observation, possibly on a time interval greater thanthat used for its estimation. Therefore, assumptions a1)–a3) arereasonable, and the model in (1) is well motivated.

As a consequence of the deterministic phase assumption a2),the process is nonstationary, i.e., its statistics change withtime. Ljung’s unified framework for deterministic plus random(thus mixed) processes is appropriate for treating single record(non)stationary processes [23, Sect. 2.3]. Let be a gener-

ally mixed process. The time-averaged expectation is denotedwith overbar and is defined as

(4)

if such a limit exists. Note that is always time-in-dependent. Two special cases of interest arise here.When is purely deterministic, the drops:

; when isstationary, time averaging has no effect: .Within this framework, we define the time-averaged autocor-relation function (ACF) of a generally complex and mixedprocess as

(5)

The natural estimator of (5) from the data is givenby

(6)

and can be shown to be asymptotically unbiased and meansquare sense consistent for the cyclostationary process (1) [7],[38].

We now show that subspace methods, such as MUSIC andESPRIT, are applicable rigorously and without approximationfor estimating the cosinusoidal frequencies in the presence ofwhite multiplicative noise. Later on, in Section IV, we showby numerical examples that they still perform well in the col-ored wideband multiplicative noise scenario so that they can beused to retrieve texture information from clutter data in the pres-ence of wideband colored speckle. However, before we proceed,we wish to point out that a harmonic-in-multiplicative-noiseprocess is different from a narrowband process, and hence, theanalysis that follows does not apply directly to the former. Anarrowband process satisfies ,where is the carrier frequency assumed to be much greaterthan the bandwidth of . At first sight, the right-hand side ofthe last equation looks like a random amplitude harmonic, butin fact, it is not, since the complex exponential is not a functionof [compare with (1)].

Consider now the real cyclostationary process of (1).The time-averaged ACF of is given by

(7)

Under the Gaussianity assumption for the speckle [6], [8], [30],[34], we obtain

(8)


where ( ) is the ACF of the ( ) component, andis the cross-correlation between theand compo-

nents. It holds true that . If the speckle iswhite noise, (8) reduces to

(9)In order to estimate , it is useful to estimate first

and remove its contribution, i.e., the line at zero frequency, fromthe data. For this purpose, define the new data as

. The mean-averaged ACF of the newdata , in the general case of colored Gaussian speckle, isgiven by

(10)

Note that the time-averaged autocorrelation in (10) has thesame structure as the autocorrelation of a stationary processcomposed by the superposition of constant amplitude har-monics with i.i.d. uniformly distributed phases and coloredadditive noise. For white speckle, i.e., white multiplicativenoise, (10) reduces to

(11)Interestingly, in this case, the time-averaged autocorrelationhas the same structure as the autocorrelation of the stationaryprocess

(12)

where the s are i.i.d. uniformly distributed in , andis white noise with variance .

It is exactly this property that allows us to employ subspacemethods that were originally proposed for estimating harmonicsembedded in additive white noise, as well as for retrieving har-monics affected by multiplicative white noise.

To proceed, define the data vector, where , and the

autocorrelation matrix . Thematrix is real, symmetric, and Toeplitz. Analogous tocommon MUSIC and ESPRIT analyses, it is easy to verifythat can be decomposed as ,whereis the so-termed steeringmatrix ,

diag , is the identity

matrix, and is the (self-) noisepower. Therefore, eigendecomposition is performed on thetime-averaged autocorrelation matrix, the same way that isapplied to constant amplitude harmonics. This shows thatwe can apply subspace methods, like MUSIC and ESPRIT,

for estimating from the clutter intensity data whenthe speckle is white. We refer to [28, Ch. 9] for subsequentdevelopment of MUSIC and ESPRIT.

Matrix is built from the estimates of for, which in turn are obtained as

(13)

where is the sample mean of the intensitydata (we do not know so we use an estimate of it). is anunbiased and mean square sense (m.s.s.) consistent estimator of

, and is an asymptotically unbiased and m.s.s. consis-tent estimator of in (11). Estimates andare related as follows:

(14)

where . As , (14)becomes .

In the next section, we will show by means of a thoroughnumerical analysis that even if MUSIC and ESPRIT have beendesigned for additive white noise and extended here to the caseof multiplicative white noise, they work fairly well also forwideband colored multiplicative noise. The performance de-pends on the “memory” of the ACF of the multiplicative noise

, that is, related to the value of the lagbeyond whichis negligible. If , then both the subspace algo-

rithms “blindly” applied to the intensity data provide accuratefrequency estimates. The intuition behind this surprisingly goodperformance is that a large-size banded Toeplitz matrix witha small “bandwidth” parameter behaves like a diagonal matrix[38].

MUSIC and ESPRIT methods provide estimates of thefrequencies . For the purpose of texture retrieval, weneed the estimates of amplitudes and phases, as well. Theterm is estimated separately as .To estimate and , we use the nonlinear leastsquares (NLLS) method, that is, we estimate the unknownparameters as the minimizers of the following criterion:

(15)

where are the frequency estimates. By making use of the no-tation found in (16) and (17), shown at the bottom of the nextpage, we can verify that the vector, which minimizes in(15), is given by

(18)

where ,i.e., is obtained from by replacing with its estimate

. Estimates and follow immediately from, and the texture is then retrieved according to (3). The fact


that the texture is non-negative was imposed in the algorithmby assuming that for those values of for which theretrieved texture is negative, but negative s never occurredin our simulations, which are described in the following section.

Remark 3: The usefulness of the theoretical results derivedin this section is not limited to the case investigated in this paper,where the multiplicative noise is common to all thecosinusoidal components. There is an abundance of literaturedealing with multicomponent models, where each componentis affected by multiplicative noise that is independent from thatof the other components (see, e.g., [9] and [24]). In this case, theobserved signal and its time-averaged autocorrelation are givenby

(19)

and

(20)

that has the same structure as the autocorrelation of a stationaryprocess composed by the superposition of constant amplitudeharmonics with i.i.d. uniformly distributed phases and coloredadditive noise. For white multiplicative noise, (20) reduces to

(21)

that has the same structure as the autocorrelation of a stationaryprocess composed by the superposition of constant amplitudeharmonics with i.i.d. uniformly distributed phases and white ad-ditive noise. Therefore, we can also employ subspace methodsfor retrieving harmonics affected by independent multiplicativewhite or wideband noises.

IV. NUMERICAL ANALYSIS OF TIME-AVERAGED

MUSIC AND ESPRIT

To evaluate the estimation accuracy of time-averaged MUSICand ESPRIT, we carried out 1000 Monte Carlo trials, generatingfor each trial a vector of complex samples according to the

model (1). The speckle is complex Gaussian distributedwith Gaussian shaped PSD. The autocorrelation sequence is

(22)

and , where is the one-lag speckle correla-tion coefficient. Note that the lower is, the shorter the length

of the multiplicative noise ACF will be. Under the above as-sumption, we find the time-averaged ACF of the new intensitydata from (10) as

(23)

The first sum on the right-hand side of (23) represents the con-tribution of the “signal” component, whereas the second onethe contribution of the “noise” component. Calculating (23) atlag zero, we obtain , whereis the term in square brackets in (23) calculated at lag, which was called self-noise in Section III. We define the

overall signal-to-noise ratio (SNR) and theth signal-to-noiseratio (SNR) as

SNR and SNR (24)

Note that since , the SNR here is alwayssmaller than 1, i.e., SNR dB. Concerning the length , ifwe define the correlation length as the value of the lag at whichthe sequence in (23) is reduced by a factor , we obtain

. For example, if , we find, whereas for , we obtain .

By Fourier transforming in (23), we obtain the time-averaged PSD of

(25)

where is a Gaussian-shaped function, symmetric around, and with standard deviation ; the

with (16)

......

...... (17)


(a)

(b)

Fig. 3. (a) Time-averaged PSD ofY (n) for � = 0:1 and 0.8. (b)Time-averaged PSD ofY (n) for � = 0.999 and 0.9999.

greater is, the narrower the multiplicative noise bandwidthwill be. Note that contains both a discrete part (2lines)and a continuous part.

In agreement with previous comments, we observe that thetime-averaged PSD in (25) has the same structure as the PSD ofa stationary process expressed as the superposition of constantamplitude harmonics with i.i.d. uniformly distributed phasesand stationary colored additive noise with PSD

. Notethat is the continuous part of , whereas the loca-tions of the lines in constitute the information we wantto retrieve. Plots of are depicted in Fig. 3(a) and (b) fordifferent values of , , , ,

, , and . Parameter affects theshape of the continuous part , which is formed by thesuperposition of terms with shape . The discretepart of is made up of lines symmetrically locatedaround zero. For very low values of , is almost flat.When increases, the shape of becomes narrower, andlikewise for . Note that the “local” SNR for each discreteline in the spectrum changes with in a way that depends

Fig. 4. Eigenvalue spectrum of the estimated correlation matrixR̂ ; K =127, SNR= �7:59 dB, SNR = SNR = SNR = �15:37 dB.

Fig. 5. Eigenvalue spectrum of the estimated modified correlation matrix;K = 127, l = 50, SNR= �7:59 dB, SNR = SNR = SNR = �15:37dB.

on the shape of as well as on the values of . InFig. 3, we observe that it decreases with; thus, we expectestimation accuracy to get worse as increases. Whenbecomes very close to one, the terms in withthe shape of become very peaky, and assumes amultimodal shape, as shown in Fig. 3(b). When , eachterm tends to a delta of Dirac and the time-averaged PSD be-comes purely discrete: The lines already present plus a lineat zero frequency. Therefore, when , behaveslike a useful signal, and estimation accuracy is expected toimprove abruptly. To numerically evaluate the performance ofthe proposed time-averaged MUSIC and ESPRIT estimators,we calculated the mean square error (MSE) for a few casestudies.

Example 1: Cosinusoids with Identical Amplitudes:We set, , , , ,

, , , , and ; asa result, we have SNR SNR SNR dB, andSNR dB. As a first step, we investigate how the rank of


Fig. 6. Modified time-averaged power spectral density;K = 127, SNR =

�7:59 dB, SNR = SNR = SNR = �15:37 dB.

Fig. 7. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of� ;K = 127,SNR= �7:59dB, SNR = SNR = SNR = �15:37 dB.

the covariance matrix depends on the one-lag speckle corre-lation coefficient . In the absence of multiplicative noise, wehave that rank . In the presence of speckle, when

, the rank is ; indeed, the completely correlated,nonzero-mean, multiplicative noise generates a line at zero fre-quency. When , we have rank , i.e.,

is full rank. In Fig. 4, we plot the first 30 eigenvalues of[estimated from a single realization through (13)] for three

different values of . It is quite evident that for ,the eigenvalues show an abrupt decrease for ;thus, for wideband speckle, can be inferred by inspectingthe estimated eigenvalue spectrum. Alternatively, (23) suggestsa way to get rid of (or reduce) the effect of multiplicative noise,therefore, obtaining a better estimate of. In fact, if we cal-culate the correlation sequence for , with

, we get . There-fore, forming the “modified”autocorrelation matrix using only the values of for lags

Fig. 8. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of the covariance matrix size;� = 0:1, SNR= �7:59 dB, SNR = SNR = SNR = �15:37 dB.

Fig. 9. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of SNR;K = 127,� = 0:1,A = 8, A = A = 3.

, we obtain a matrix with rank . This suggestsestimating by counting the number of dominant eigenvaluesin the eigenvalue spectrum of the modified . In Fig. 5, weplot the first 30 eigenvalues of the modified . Alternatively,we may count the dominant (and symmetric) peaks in the mod-ified time-averaged PSD obtained by Fourier transforming thesequence . In Fig. 6, we plot one esti-mate of the modified time-averaged PSD for ; when

, the six symmetric peaks are even more distinguish-able. The curves in Figs. 5 and 6 were obtained for and

. Note that ; in fact, for , we found. Finally, it is worth stressing that since the time-aver-

aged ACF in (10) and (23) has the same structure as the auto-correlation of a stationary process composed of the superposi-tion of constant amplitude harmonics with i.i.d. uniformly dis-tributed phases and colored additive noise, all the methods formodel order determination already proposed for MUSIC in col-ored noise can be applied as well.


Fig. 10. Mean square amplitude estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of� ;K = 127, SNR= �7:59

dB, SNR = SNR = SNR = �15:37 dB.

Fig. 11. Texture retrieved by means of MUSIC and ESPRIT;K = 127,� =

0:1, SNR= �7:59 dB, SNR = SNR = SNR = �15:37 dB.

All the numerical results presented in the following were de-rived by applying MUSIC and ESPRIT as described in [28, Ch.9] to the unmodified covariance matrix . However, an inter-esting development of the present work will be to investigatethe performance of subspace methods applied to the modified

. In Fig. 7, we report the MSE of the frequency estimatesas a function of for both MUSIC and ESPRIT methods.Lines without marks refer to ESPRIT and lines with marks ()to MUSIC. For , the two methods have almost thesame estimation accuracy. We also investigated the performancefor , but the results are not reported here for lack ofspace. In this case, ESPRIT exhibits better performance thanMUSIC for almost every value of . Note that in Fig. 7, thelower is, the better the estimation accuracy will be, but when

, estimation accuracy improves abruptly. The MSE be-havior as a function of also depends on , on the SNRs,and on , e.g., for and the other parameters as inFig. 7, we found that the MSE is almost constant for(it decreases very slowly) and decreases abruptly when .

This abrupt decrease is also present for . The differentbehavior of the MSE versus for and isprobably due to the following reason. It is known that the accu-racy of MUSIC and ESPRIT frequency estimates increases withincreasing (remember that is the size of the covariancematrix ); hence, this value should be chosen as large as pos-sible, but not too close to , in order to obtain a reliable esti-mation of . The variance of increases both with thelag and with the value of (because the number of “effec-tive” independent samples decreases with). Whenand , the estimates for the largest values of

are quite good, and thus, the effect of on their variance isnegligible; therefore, MSE is almost constant. Forand , estimates of for the largest values ofsuffer for non-negligible variance, and this variance gets worsewhen increases; thus, MSE also gets worse with .

To further investigate the effect of on frequency estima-tion accuracy, we derived the results plotted in Fig. 8, whereincreases from 15 to 255, which is lower than . Weobserve that for small , ESPRIT performs better than MUSIC,whereas for large , the situation is reversed (but for ,they have the same performance). Moreover, it is worth ob-serving that the MSE decreases quite abruptly whengoesfrom 16 to 64 and more slowly after that. In fact, for ,the correlation length of the speckle is , and theestimation algorithms perform like in additive white noise.

In Fig. 9, we allowed to vary while keeping andconstant, leading to variable SNR, SNR , and SNR. Indeed,from (24), we have that SNR .When SNR dB, the three SNRs are equal. Note thatthe range of variation of SNRhas be chosen such that

is satisfied for every . In this range, MUSIC outperformsESPRIT.

Curves of the MSE for the amplitude estimators in (18) areplotted in Fig. 10. The two methods exhibit almost identicalperformance (MUSIC is slightly better). Similar results werefound for the phases. The estimates of , , and

are then plugged into (3) to retrieve texture information.In Fig. 11, we show the true , which is the same as in Figs. 1and 2, and obtained by means of ESPRIT and MUSIC.The improvement over the nonparametric WP estimator of (2)is quite evident. To quantify it, we calculate the time-averagedexpectation of the squared error of the texture estimate de-fined as

(26)

In practice, we estimated by first calculatingfor each Monte Carlo (MC)

run and then averaging over the 1000 MC runs. Wefound , , and

for , for , andfor (see also Fig. 2). This confirms previous results.

Thus far, we have assumed that the value ofis knownor that it is perfectly estimated from the eigenvalue spectrumof . To investigate how a wrong estimate of affects theaccuracy of texture retrieval, we assumed that the number ofcomponents in is , whereas the actualnumber is still . For each , we estimated running


Fig. 12. Time-averaged MSE of̂�(n) retrieved by means of MUSIC andESPRIT as a function of the estimated value ofM .

1000 Monte Carlo simulations. The results, which are plottedin Fig. 12, show that the lower is, the lower will be (asexpected) and, in addition, the lower the robustness to a wrongestimate of will also be. However, the results also show thatan increase in is more tolerable if is overestimated ratherthan underestimated. In fact, it is quite natural, e.g., , toexpect . The error will be much greater if , andas a consequence, we erroneously set . However, it isworth stressing that results in Fig. 12 refer to a scenario whereall the components have the same amplitude, which is a sort ofworst-case scenario. In a realistic radar application, we expectthe amplitudes to decrease with the order[18] andthe method to be quite robust even to non-negligible errors on

. We further elaborate on this problem in the next example.Example 2: Cosinusoids with Different Amplitudes:We set

, , , , , and ,, , , and .

As a result, we have SNR dB, SNR dB,SNR dB, and SNR dB. Plots in Fig. 13show that MUSIC outperforms ESPRIT for the frequency withthe highest SNR, but for the other two frequencies, they have al-most the same accuracy. Plots in Fig. 14 show that increasingimproves estimation accuracy only for the line with the highestSNR. The MSE of the other two estimates is scarcely sensi-tive to changes of . Amplitudes and phases are estimated ac-cording to (18). The retrieved texture information is plotted inFig. 15 for and . In this case, we found

, , for, for , and for .

To further investigate how a wrong estimate ofaffects theaccuracy of texture retrieval, we generated a more realistic tex-ture, composed of components with different ampli-tudes,2 and we repeated the simulations (#MC1000) describedin Example 1to derive the results plotted in Fig. 12. The data sizeis still . In Fig. 16, we plot the amplitudesas a function of frequency, with the locations of the lines being

2The frequencies, amplitudes, and phases of�(n) used in this example wereobtained by processing, through the method based on time-averaged ESPRIT,the measured high-resolution sea clutter data described in [8] and [18].

Fig. 13. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of� ; components with differentSNRs,K = 127.

Fig. 14. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of the matrix sizeK; componentswith different SNRs� = 0:1.

Fig. 15. Texture retrieved by means of MUSIC and ESPRIT; components withdifferent SNRs,K = 127, � = 0:1.


Fig. 16. AmplitudesfA g versus frequencyM = 8.

Fig. 17. Texture versus timeM = 8.

determined by the values of . The actual behavior ofthe texture is plotted in Fig. 17 (only the first 300 sam-ples). The results shown in Fig. 18 confirm that when the methodworks well (i.e, when is small), it is also quite robust to er-rors in the estimation of the number of components. In fact,in this case, the -curve is quite flat around the min-imum. Interestingly, for large , the curve is monotonicallyincreasing with , in the range we have investigated. However,in this case, the error is quite large for as well. Thisconfirms that subspace methods can be employed to solve suchkinds of estimation problems only when the multiplicative noiseis white or colored wideband, in agreement with the theoreticalresults of the previous section.

Example 3: Two Closely Spaced Cosinusoids:As is wellknown, MUSIC and ESPRIT are super-resolution frequencyestimation algorithms, that is, they guarantee better res-olution than , which is the so-called Rayleigh limit.This characteristic is fundamental for sea clutter analysispurposes. Therefore, to complete the analysis of the twoalgorithms, we evaluate the MSE of the two methods whenestimating two frequency lines spaced apart less than.To this purpose, we generated 1000 random sequences

Fig. 18. Time-averaged MSE of̂�(n) retrieved by means of ESPRIT as afunction of the estimated value ofM ; K = 127, M = 8.

Fig. 19. Periodogram ofY (n): f = 0:04, f = 0:0406,� = 0:1, SNR=�6:025 dB; SNR = SNR = �12:05 dB.

of samples so that .Then, we set , , (so thatSNR SNR SNR dB),

, , and . The speckle processis generated as in Examples 1 and 2. Fig. 19 shows the peri-odogram of one data sequence in the range .It is clear that the two frequency lines are indistinguishable,not only because , butalso because of the presence of the wideband multiplicativenoise which smears the periodogram. This is not a surprisebecause we stated at the very beginning that due to the presenceof wideband multiplicative noise, the frequencies cannot beestimated simply from the periodogram of the intensity data,which amounts to calculating the FFT of and locatingthe 2 highest peaks. However, (23) and (25) and Fig. 3suggest that to estimate the frequencies, we can first estimatethe time-averaged ACF and then locate the 2 highestpeaks in the FFT of , which represents an FFT-based es-timate of . From (25), where the lines appear unsmeared,we have that the best resolution we can obtain with this method


Fig. 20. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of the speckle one-lag correlationcoefficient;K = 127.

Fig. 21. Mean square frequency estimation error by means of ESPRIT (lines)and MUSIC (lines with marks�) as a function of the covariance matrix size;� = 0:1.

is . Therefore, the well-known Rayleigh limit holds validin this case as well, provided that we refer to the time-averagedPSD rather than to the periodogram of the intensity data. It isexactly this fact that motivates the use of subspace methods forestimating closely spaced frequencies in the presence of whiteor wideband colored multiplicative noise. The main issue in ourapplication is the parametric estimate of a slowly varying signaland the texture, which may contain low and closely spacedfrequencies [17], [18]. The better the estimates of frequencies,amplitudes, and phases, the better the texture retrieval will be.

To evaluate numerically the performance, we calculated theMSE on frequency estimation as a function of (Fig. 20) and

(Fig. 21). Comparing these results with those shown in Figs. 7and 8, we observe that the MSEs for closely spaced frequen-cies are considerably higher. In this case, is not suffi-cient to guarantee good performance, and we have to use highervalues of at the cost of increasing computational complexity.For example, if we require MSE from Fig. 6, werequire , and for closely spaced harmonics, we need

. Plots of the MSE for amplitude and phase estimatesare not reported here due to lack of space.

V. CONCLUDING REMARKS

The main contribution of this paper is to show that theproblem of retrieving harmonics observed in white (or wide-band colored) multiplicative noise can be carried out usingsimilar tools as in retrieving constant amplitude harmonicsobserved in white (or wideband colored) additive noise. Thisresult was reached by employing Ljung’s time-averagedframework for nonstationary processes and, in particular, theconcept of time-averaged autocorrelation function and powerspectral density. Eigendecomposition was performed on thetime-averaged autocorrelation matrix, the same way that itis applied to constant amplitude harmonics. This allowed usto establish that MUSIC and ESPRIT are also applicable toharmonics observed in white wideband colored multiplicativenoise. We then applied the time-averaged subspace estimatorsto solve the problem of retrieving the texture of high-resolutionand low-grazing angle sea clutter. The texture was modeled asa cyclostationary signal, taking into account the periodic struc-ture of the sea surface. In this application, the multiplicativenoise models the square of the absolute value of the clutterspeckle samples. It is colored wideband with unknown spectra,nonzero mean, and non-Gaussian distributed. Simulationexamples illustrated that subspace algorithms can be appliedto retrieve texture information without requiring knowledgeof the speckle spectral shape. Numerical results suggest thatgenerally MUSIC slightly outperforms ESPRIT, but both ofthem enable separation of closely spaced harmonics present inthe texture when classical FFT based methods cannot, at theprice of increased computational complexity (large), andprovide better texture retrieval performance than standard non-parametric methods. The problems of estimating the numberof cosinusoidal components in the texture and the effects ontexture retrieval of a wrong estimate were also considered. Thenext step in our research is the validation of subspace methodperformance prediction against real recorded sea clutter data.

REFERENCES

[1] N. Allan, D. B. Trizna, and D. J. McLaughlin, “Numerical comparisonof techniques for estimating doppler velocity time series from coherentsea surface scattering measurement,”Proc. Inst. Elect. Eng. F, vol. 145,pp. 367–373, Dec. 1998.

[2] V. Anastassopoulos, G. A. Lampropoulos, A. Drosopoulos, and M. Rey,“High resolution radar clutter statistics,”IEEE Trans. Aerosp. Electron.Syst., vol. 35, pp. 43–60, Jan. 1999.

[3] B. C. Armstrong and H. D. Griffiths, “Modeling spatially correlatedK-distributed clutter,”Electron. Lett., vol. 27, no. 15, pp. 1355–1356,July 18, 1991.

[4] O. Besson, F. Gini, H. D. Griffiths, and F. Lombardini, “Estimatingocean surface velocity and coherence time using multichannel ATI-SARsystems,”Proc. Inst. Elect. Eng. F, vol. 147, no. 6, pp. 299–308, Dec.2000.

[5] H. C. Chan, “Radar sea-clutter at low grazing angles,”Proc. Inst. Elect.Eng. F, vol. 137, no. 2, pp. 102–112, Apr. 1990.

[6] E. Conte and M. Longo, “Characterization of radar clutter as a spheri-cally invariant random process,”Proc. Inst. Elect. Eng. F, vol. 134, no.2, pp. 191–197, Apr. 1987.

[7] A. V. Dandawate and G. B. Giannakis, “Asymptotic theory of mixedtime averages andkth-order cyclic-moment and cumulant statistics,”IEEE Trans. Inform. Theory, vol. 41, pp. 216–232, Jan. 1995.


[8] A. Farina, F. Gini, M. V. Greco, and L. Verrazzani, “High resolution seaclutter data: A statistical analysis of recorded live data,”Proc. Inst. Elect.Eng. F, vol. 144, no. 3, pp. 121–130, June 1997.

[9] G. B. Giannakis and G. Zhou, “Harmonics in multiplicative and additivenoise: Parameter estimation using cyclic statistics,”IEEE Trans. SignalProcessing, vol. 43, pp. 2217–2221, Sept. 1995.

[10] F. Gini, M. Luise, and R. Reggiannini, “Analysis and design of a DPSKoptical heterodyne receiver in the presence of laser phase noise and fre-quency detuning,” inInt. J. Commun. Syst., 1995, vol. 8, pp. 129–141.

[11] F. Gini, “Sub-optimum coherent radar detection in a mixture ofK-dis-tributed and Gaussian clutter,”Proc. Inst. Elect. Eng. F, vol. 144, no. 1,pp. 39–48, Feb. 1997.

[12] F. Gini and G. B. Giannakis, “Frequency offset and symbol timing re-covery in flat fading channels: A cyclostationary approach,”IEEE Trans.Commun., vol. 46, pp. 400–411, Mar. 1998.

[13] F. Gini, M. Luise, and R. Reggiannini, “Cramér–Rao bounds in the para-metric estimation of fading radio transmission channels,”IEEE Trans.Commun., vol. 46, pp. 1390–1398, Oct. 1998.

[14] F. Gini, M. V. Greco, A. Farina, and P. Lombardo, “Optimum and mis-matched detection againstK-distributed plus Gaussian clutter,”IEEETrans. Aerosp. Electron. Syst., vol. 34, pp. 860–876, July 1998.

[15] F. Gini and M. V. Greco, “Suboptimum approach to adaptive coherentradar detection in compound-Gaussian clutter,”IEEE Trans. Aerosp.Electron. Syst., vol. 35, pp. 1095–1104, July 1999.

[16] F. Gini, M. Greco, M. Diani, and L. Verrazzani, “Performance analysisof adaptive radar detectors against non-Gaussian real sea clutter data,”IEEE Trans. Aerosp. Electron. Syst., vol. 36, pp. 1429–1439, Oct. 2000.

[17] F. Gini and M. Greco, “Texture modeling and validation using measuredsea clutter data,” in2001 IEEE Radar Conf., Atlanta, GA, May 2001, pp.387–392.

[18] , (2001, June) Texture modeling, estimation, and val-idation using measured sea clutter data [Online] . Available:http://diani.iet.unipi.it/Gini/download.html

[19] T. Hair, T. Lee, and C. J. Baker, “Statistical properties of multifrequencyhigh-range-resolution sea reflections,”Proc. Inst. Elect. Eng. F, vol.138, no. 2, pp. 75–79, Apr. 1991.

[20] F. Henderson and A. Lewis, “Principle & applications of imaging radar,”in Manual of Remote Sensing. New York: Wiley, 1998, vol. 2.

[21] S. Haykin and D. J. Thomson, “Signal detection in a nonstationary en-vironment reformulated as an adaptive pattern classification problem,”Proc. IEEE, vol. 86, pp. 2325–2344, Nov. 1998.

[22] D. E. Kreithen, S. M. Crooks, W. W. Irving, and S. D. Halversen, “Es-timation and detection using the product model,” Lincoln Lab., Mass.Inst. Techol., Lexington, Project Rep. STD-37, 1991.

[23] L. Ljung, System Identification: Theory for the User. EnglewoodCliffs, NJ: Prentice-Hall, 1987.

[24] F. Lombardini, F. Gini, and P. Matteucci, “Application of array pro-cessing techniques to multibaseline InSAR for layover solution,” in2001IEEE Radar Conf., Atlanta, GA, May 2001, pp. 210–215.

[25] P. Lombardo and A. Farina, “Coherent radar detection againstK-dis-tributed clutter with partially correlated texture,”Signal Process., vol.48, pp. 1–15, Jan. 1996.

[26] T. J. Nohara and S. Haykin, “Canadian East Coast radar trials and theK-distribution,” Proc. Inst. Elect. Eng. F, vol. 138, no. 2, pp. 80–91,Apr. 1991.

[27] W. J. Plant and W. C. Keller, “Evidence of Bragg scattering in microwavedoppler spectra of sea return,”J. Geophys. Res., vol. 95, no. C9, pp.16 299–16 310, Sept. 15, 1990.

[28] B. Porat, Digital Processing of Random Signals, Theory &Methods. Englewood Cliffs, NJ: Prentice-Hall, 1994.

[29] F. L. Posner, “Experimental observations at very low grazing angles ofhigh range resolution microwave backscatter from the sea,”, Naval Res.Lab. Rep. NRL/MR/5310-98-8326, Dec. 31, 1998.

[30] M. Rangaswamy, D. D. Weiner, and A. Ozturk, “Non-Gaussian vectoridentification using spherically invariant random processes,”IEEETrans. Aerosp. Electron. Syst., vol. 29, pp. 111–124, Jan. 1993.

[31] M. Rangaswamy, J. H. Michels, and D. D. Weiner, “Multichannel detec-tion algorithm for correlated non-Gaussian random processes based oninnovations,”IEEE Trans. Signal Processing, vol. 43, pp. 1915–1922,Aug. 1995.

[32] C. L. Rino, E. Eckert, A. Siegel, T. Webster, A. Ochadlick, M. Rankin,and J. Davis, “X-Band low-grazing-angle ocean backscatter obtainedduring LOGAN 1993,”IEEE J. Oceanic Eng., vol. 22, pp. 18–26, Jan.1997.

[33] E. Rodriguez and J. M. Martin, “Theory and design of interferometricsynthetic aperture radars,”Proc. Inst. Elect. Eng. F, vol. 139, no. 2, pp.147–159, Apr. 1992.

[34] K. J. Sangston and K. R. Gerlach, “Coherent detection of radar targets ina non-Gaussian background,”IEEE Trans. Aerosp. Electron. Syst., vol.30, pp. 330–340, Apr. 1994.

[35] K. J. Sangston, F. Gini, M. V. Greco, and A. Farina, “Structures for radardetection in compound Gaussian clutter,”IEEE Trans. Aerosp. Electron.Syst., vol. 35, pp. 445–458, Apr. 1999.

[36] K. D. Ward, C. J. Baker, and S. Watts, “Maritime surveillanceradar—Part 1: Radar scattering from the ocean surface,”Proc. Inst.Elect. Eng. F, vol. 137, pp. 51–62, Apr. 1990.

[37] G. Zhou and G. B. Giannakis, “On estimating random amplitude modu-lated harmonics using higher-order spectra,”IEEE J. Oceanic Eng., vol.19, pp. 529–539, Oct. 1994.

[38] G. Zhou, “Random amplitude and polynomial phase modeling of non-stationary processes,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Vir-ginia, Charlottesville, Jan. 1995.

Fulvio Gini (SM’00) received the Dr.Eng. (cumlaude) and the Research Doctor degrees in electronicengineering from the University of Pisa, Pisa, Italy,in 1990 and 1995, respectively.

In 1993, he joined the Department of “Ingegneriadell’Informazione” of the University of Pisa, wherehe has been an Associate Professor since October2000. From July 1996 through January 1997, hewas a Visiting Researcher with the Departmentof Electrical Engineering, University of Virginia,Charlottesville. He has given lectures at universities

in Italy and abroad and gave a tutorial entitled “Coherent Detection and Fusionin High Resolution Radar Systems” at the International Conference on Radar(May 1999, Brest, Germany) and the tutorial “Advanced Radar Detectionand Fusion” at the International Radar Conference (May 2000, WashingtonD.C.). His general interests are in the areas of statistical signal processing,estimation, and detection theory. In particular, his research interests includenon-Gaussian signal detection and estimation using higher order statistics,parameter estimation and data extraction from multichannel interferometricSAR data, cyclostationary signal analysis, and estimation of nonstationarysignals, with applications to communication and radar processing. He is theauthor of about 50 journal papers and 40 conference papers.

Dr. Gini is an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL

PROCESSING. He has been session chairman at international conferences.

Georgios B. Giannakis(F’97) received the Diploma in electrical engineeringfrom the National Technical University of Athens, Athens, Greece, 1981. FromSeptember 1982 to July 1986, he was with the University of Southern California(USC), where he received the M.Sc. degree in electrical engineering in 1983,the M.Sc. degree in mathematics in 1986, and the Ph.D. degree in electricalengineering in 1986.

After lecturing for one year at USC, he joined the University of Virginia,Charlottesville, in 1987, where he became a Professor of electrical engineeringin 1997. Since 1999, he has been with the University of Minnesota, Minneapolis,as a Professor of electrical and computer engineering. His general interests spanthe areas of communications and signal processing, estimation and detectiontheory, time-series analysis, and system identification—subjects on which hehas published more than 120 journal papers, 250 conference papers, and twoedited books. Current research topics focus on transmitter and receiver diversitytechniques for single- and multiuser fading communication channels, redundantprecoding and space-time coding for block transmissions, and multicarrier andwideband wireless communication systems. He is a frequent consultant for thetelecommunications industry.

Dr. Giannakis was the (co-) recipient of three best paper awards from theIEEE Signal Processing (SP) Society in 1992, 1998, and 2000. He also re-ceived the Society’s Technical Achievement Award in 2000. He co-organizedthree IEEE-SP Workshops (HOS in 1993, SSAP in 1996, and SPAWC in 1997)and guest (co-) edited four special issues. He has served as an Associate Ed-itor for the IEEE TRANSACTIONS ONSIGNAL PROCESSINGand the IEEE SIGNAL

PROCESSINGLETTERS, a secretary of the SP Conference Board, a member of theSP Publications Board, and a member and vice chair of the Statistical Signaland Array Processing Committee. He is a member of the Editorial Board forthe PROCEEDINGS OF THEIEEE, chairs the SP for Communications TechnicalCommittee, and serves as the Editor in Chief for the IEEE SIGNAL PROCESSING

LETTERS. He is a member of the IEEE Fellows Election Committee and theIEEE-SP Society’s Board of Governors.


Maria Greco (M’98) was born in Lecce, Italy, in 1968. She received the B.S.degree in electronic engineering in 1993 and the Ph.D. degree in 1998 fromUniversity of Pisa, Pisa, Italy.

From December 1997 to May 1998, she joined the Georgia Institute of Tech-nology, Atlanta, as a Visiting Research Scholar, where she carried on researchin the field of radar detection in a non-Gaussian background. In 1993, she joinedas the Department of “Ingegneria dell’Informazione,” University of Pisa, whereshe has been an Assistant Professor since April 2001. Her main interests includecyclostationary signal analysis, clutter models, coherent and incoherent detec-tion in non-Gaussian clutter, and CFAR techniques.

G. Tong Zhou (SM’00) received the B.Sc. degree in biomedical engineeringand instrumentation from the Tianjin University, Tianjin, China, in July 1989.From September 1989 to May 1995, she was with the University of Virginia(UVA), Charlottesville, where she received the M.Sc. degree in biophysics inMay 1992, the M.Sc. degree in electrical engineering in January 1993, and thePh.D. degree in electrical engineering in January 1995.

She has been with the School of Electrical and Computer Engineering,Georgia Institute of Technology (Georgia Tech), Atlanta, since September1995 and currently holds the rank of Associate Professor. Her research interestsare in the general areas of statistical signal processing and communications.Specific current interests include nonlinear system identification, predistortionlinearization, blind deconvolution, network traffic analysis, and biosignalanalysis.

Dr. Zhou was awarded the 1995 Allan Talbott Gwathmey Memorial Awardfor outstanding research in the physical sciences at UVA based on her Ph.D. dis-sertation. In 1997, she received the National Science Foundation Faculty EarlyCareer Development (CAREER) Award. She also received the 2000 MeritorTeaching Excellence Award at Georgia Tech. She is a member of Eta KappaNu.

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