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Radar Detection Schemes for Joint Temporal andSpatial Correlated Clutter Using Vector ARMA Models

Wajih Ben Abdallah, Jean-Philippe Ovarlez, Pascal Bondon

To cite this version:Wajih Ben Abdallah, Jean-Philippe Ovarlez, Pascal Bondon. Radar Detection Schemes for Joint Tem-poral and Spatial Correlated Clutter Using Vector ARMA Models. 25th European Signal ProcessingConference (EUSIPCO 2017), Aug 2017, Kos island, Greece. 5 p., �10.23919/eusipco.2017.8081373�.�hal-01578366�

Radar Detection Schemes for Joint Temporal andSpatial Correlated Clutter Using Vector ARMA

ModelsWajih Ben Abdallah

L2S and SONDRA, CentraleSupelecGif-sur-Yvette, France

Jean Philippe OvarlezSONDRA and French Aerospace Lab

ONERA DEMR/TSI, Palaiseau, France

Pascal BondonL2S, CNRS

Gif-sur-Yvette, France

Abstract—Adaptive radar detection and estimation schemesare often based on the independence of the training data usedfor building estimators and detectors. This paper relaxes thisconstraint and deals with the non-trivial problem of derivingdetection and estimation schemes for joint spatial and temporalcorrelated radar measurements. In order to estimate these twojoint correlation matrices, we propose to use the Vector ARMA(VARMA) methodology. The estimation of the VARMA modelparameters are performed with Maximum Likelihood Estimatorsin Gaussian and non-Gaussian environment. These two jointestimates of the spatial and temporal covariance matrices leadsto build Adaptive Radar Detectors, like Adaptive NormalizedMatched Filter (ANMF). Their corresponding performance areanalyzed through simulated datasets. We show that taking intoaccount the spatial covariance matrix may lead to significantperformance improvements compared to classical proceduresignoring the spatial correlation.

I. INTRODUCTION

In many applications, data can be viewed as a joint spatialand temporal process. For example, in meteorology sets ofrainfall measurements can be collected from a number ofdifferent sites over a period of time [1], [2]. In radar andimagery applications, this view point can be of high interest.For high resolution radar, the sea clutter is clearly jointlyspatially and temporally correlated [3]. In multichannel (po-larimetric, interferometric or multi-temporal) SAR imaging,the multivariate vector characterizing each spatial pixel of theimage is correlated over the channels but can also be stronglycorrelated with those of neighbourhood pixels. Finally, theneed of simulating doubly correlated temporal and spatialprocess is of importance. In the radar community, one gen-erally supposes that the vectors of information collected overa spatial support are identically and independently distributed(IID). This strong assumption allows to build easily Maxi-mum Likelihood Estimators of parameters like for example,covariance matrix required for adaptive detection, leading tooverestimated performance of such detectors. The aim of thispaper is to relax this hypothesis through the use of space-time autoregressive and moving average vector models in thecontext of wide radar applications.

The vector ARMA (VARMA) model has been widely usedfor modeling the temporal or spatial dependence structureof a multivariate discrete signal. Unlike scalar signals, thetemporal or spatial dependence of a vector signal consists of

not only the serial dependence within each marginal signal,but also the interdependence across different marginal signals.The VARMA model is well suited to represent such temporaldependence structures. More precisely, (yk)k∈Z is an m-variate zero-mean VARMA(p, q) model if (yk) satisfies thedifference equation

yk −p∑

i=1

Φiyk−i = ck +

q∑i=1

Θick−i, (1)

where Φ1, . . . ,Φp,Θ1, . . . ,Θq are real or complex m × mmatrices and (ck) is a sequence of IID m-variate zero-meanvectors with non-degenerate covariance matrix Σc charac-terizing the temporal dependence of its components. Thereare exisiting works that concern only the autoregressive (AR)modelisation of the spatio-temporal correlations between theclutter [4], [5], [6]. This implies simplifications since theCMLE has a closed form expression in this case. However,the AR model does not necessarily reflect the true correlationstructure and it is interesting to study the dual model whichis the MA model. The process (ck) associated to radar clutterreturns for example can be non-Gaussian and have for instancea Complex Elliptically Symmetric (CES) distribution [7]. Thematrices Φi and Θi satisfies some constraints to guaranty theexistence, uniqueness, causality and invertibility of a solution(yk) to (1), see e.g. [8]. The estimation of gaussian VARMAmodels in the space domain by maximizing the likelihoodand in the space-frequency domain by maximizing the socalled Whittle’s approximation to the Gaussian likelihood isconsidered by [9] and [10], see also [11]. In radar detectionschemes, covariance or scatter matrix estimation (for CESdistributed clutter) is a fundamental problem. To estimate thescatter matrix of any observed vector under test y (primaryvector on dimension m), generally it is supposed to disposeof K > m secondary IID vectors (training data) that sharewith the primary vector the same statistical characteristics [12],[13]. In some applications, as in high spatial resolution radars,the hypothesis of independence (or even of uncorrelation) ofthe secondary vectors is seldom satisfied due to the nature ofthe phenomenon at hand. Particularly, for target detection pur-poses, due to the lack of knowledge of the spectral characteris-tic of the clutter and the variability of it on long periods of time

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

M SCM

MG

CML

MG

EML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

M SCM

MG

CML

MG

EML

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

M SCM

MG

CML

MG

EML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

M SCM

MG

CML

MG

EML

(a) (b) (c) (d)

Fig. 1. Pfa − λ plot and the corresponding Pd-SNR relationship for Pfa = 10−2 for spatially non correlated Gaussian clutter (m = 16, ρ = 0.5). (a), (b)K = 32. (c), (d) K = 48.

and large surfaces, the covariance matrix of the clutter mustbe estimated and plugged into adaptive detectors in both casesof Gaussian (Adaptive Matched Filter, Kelly detector) andnon-Gaussian distributed disturbance (Adaptive NormalizedMatched Filter). Recent works have also considered spatialcorrelation [14], [15], [16].

In this paper we assume that the clutter is jointly tem-porally and spatially correlated. These correlations are mod-eled through a VARMA process for both Gaussian and nonGaussian signals [17]. Solving this problem can improve theperformance of detection. We propose to use the well-knownGaussian Maximum Likelihood (ML) estimator [17] and weextend this formulation for non Gaussian ML formulationin the case of CES innovations. This paper is organizedas follows. Section 2 gives a summary of radar detectionbackground and states the problem. Section 3 is dedicatedto the estimation of the parameters in the Gaussian case.Section 4 presents the detailed steps of Maximum LikelihoodEstimators (MLE) formulation in the non Gaussian case. Sec-tion 5 gives some results and discussions about the effect of thespatial and temporal clutter correlations on the radar detectionperformances through simulated datasets. Conclusions andperspectives are presented in section 6.

II. RADAR DETECTION BACKGROUND

In radar detection theory, the detection problem can bestated as a classical binary hypothesis testing. A receivedcomplex clutter observation y is considered as a target-freesignal (H0 hypothesis) or a signal containing the target (H1

hypothesis) as follows,{H0 : y = c yk = ck, k = 1, . . . ,KH1 : y = αp + c yk = ck, k = 1, . . . ,K

, (2)

where y is the m-vector observed signal, ck is complexsecondary data, assumed to be independent, α is the unknowncomplex target amplitude and p denotes a known steeringvector [18]. The goal of the detection theory is to decidebetween these two hypotheses and the corresponding detectiontest performance is analyzed through the false alarm proba-bility Pfa and the probability of detection Pd behaviors. Aswe propose in this paper to compare performances both inGaussian and non-Gaussian clutters, we will focus on thewell known Normalized Matched Filter (GLRT in partially

homogeneous Gaussian clutter [19], approximated GLRT innon-Gaussian CES clutter [20]) defined by

ΛANMF =

∣∣∣pHM−1

y∣∣∣2(

pHM−1

p)(

yHM−1

y) H1

≷H0

λ, (3)

where λ is the detection threshold and where M stands forany estimator of the covariance matrix M of y. Due tothe joint temporal and spatial correlations characterizing theclutter, this estimation is crucial for improving the detectionperformance. For this reason we attempt in this paper to use asimple adequate time series model that takes into account thespatial correlation of the observed data in order to improvethe estimation accuracy. The chosen model is a VARMA(0,1),i.e. each secondary data is correlated only with the previoussecondary data, which leads to the following model where ckis the innovation and Θ1 is a matrix coefficient,

yk = ck + Θ1ck−1, k = 1, . . . ,K. (4)

In the following, we propose to analyze the effect of varyingthe data distribution structure (independent or spatially corre-lated) on the radar detection performance under Gaussian andnon-Gaussian assumptions. Since the sequence (ck) is zero-mean and IID, the covariance matrix M = E(yky

Hk ) in (4)

satisfies

M = E(ckcHk ) + Θ1E(ckc

Hk )ΘH

1 = Σc + Θ1ΣcΘH1 . (5)

III. GAUSSIAN CASE

Here, we assume that the observations yk follow a Gaussiandistribution and we analyze the two cases where the data areeither independent or follow a VARMA(0,1) model.

A. Independent observations

For the rest of this paper, let Y =(yT1 ,y

T2 , . . . ,y

TK

)T. For

independent Gaussian distributed data, the likelihood functionL1(Y ,M) is given by

L1(Y ;M) =1

πmK |M |Kexp

(−

K∑k=1

yHk M−1yk

), (6)

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

M SCM

MG

CML

MG

EML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

M SCM

MG

CML

MG

EML

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

M SCM

MG

CML

MG

EML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

M SCM

MG

CML

MG

EML

(a) (b) (c) (d)

Fig. 2. Pfa −λ plot and the corresponding Pd-SNR relationship for Pfa = 10−2 for spatially VARMA(0,1) correlated Gaussian clutter (m = 16, ρ = 0.5,Θ1 = 0.9Im). (a), (b) K = 32. (c), (d) K = 48.

where |M | = det(M). The maximum of L1(Y ;M) withrespect to M is the well-known Sample Covariance Matrix(SCM) MSCM given by

MSCM =1

K

K∑k=1

ykyHk . (7)

B. VARMA(0,1) observations

The VARMA(0,1) model (4) is specified by the covariancematrix Σc of the innovations ck and the coefficients matrixΘ1. The estimation techniques of these parameters are basedon the conditional or the exact MLE [17].

1) Conditional likelihood function: The conditional MLEis obtained by assuming that c0 = 0. It follows from (4) thatC =

(cT1 , c

T2 , . . . , c

TK

)Tis related to Y by C = TY where

T =

Im 0m . . . 0m

−Θ1 Im . . . 0m

(−Θ1)2 −Θ1 . . . 0m

......

. . ....

(−Θ1)K−1 (−Θ1)K−2 . . . Im

. (8)

Since the innovations ck are IID and det(T ) = 1, theconditional likelihood function of Y is

L2(Y ; Θ1,Σc) =1

πmK |Σc|Kexp

(−

K∑k=1

cHk Σ−1c ck

). (9)

We denote by ΘG

1,CML and ΣG

c,CML the values of Θ1 and Σc

which maximize (9), and we denote by MG

CML the value ofM obtained by replacing respectively Θ1 and Σc by Θ

G

1,CML

and ΣG

c,CML in (5).2) Exact likelihood function: The different steps in com-

puting the exact likelihood function are similar to the onesdescribed in the conditional case except that we must estimatethe value of the initial innovation c0. Similarly to the con-ditional case, we observe that the determinant of the matrixrelating the observations to the innovations is equal to one,and then the joint Probability Density Function (PDF) of theobservations yk is equal to the product of the PDF’s of theinnovations, i.e.,

p(c0,y1,y2, . . . ,yK) =

K∏k=0

p(ck). (10)

The exact likelihood function L3(Y ; Θ1,Σc) is obtained byintegrating (10) out c0. It was shown in [17] that

L3(Y ; Θ1,Σc) =exp (−(Y − Xc0)H(Y − Xc0))

π|Σc|K |XHX|

, (11)

where c0, X and Y depend only on the observations and Σc

and Θ1. We denote by ΘG

1,EML and ΣG

c,EML the values of Θ1

and Σc that maximize (11). From these estimated matrices wecompute M

G

EML using (5). The exact procedure is particularlycomputationally expensive to solve when Σc is not structured(m × (m + 1)/2 elements to estimate). In this paper, wehave solved the problem when Σc has a Toeplitz structuredepending on only one parameter.

As the sample size tends to infinity, the conditional MLE’sΘ

G

1,CML and ΣG

c,CML, and the exact MLE’s ΘG

1,EML and

ΣG

c,EML both converge almost surely to the true values Θ1

and Σc. But their finite sample behaviours may be different,especially in the case of small datasets.

IV. NON GAUSSIAN CASE

In real measurements, the clutter is often non-Gaussiandistributed. One of the main class of these non Gaussianstatistical models is the compound Gaussian processes (CESsubclass). The random vector ck is compound Gaussian ifck =

√τkxk where τk is a positive random variable called

texture which measures the variability of the local power ofthe received signal, and xk is a m-dimensional zero-meanGaussian vectors with covariance matrix Σx called speckle,and τk and xk are independent. We assume in the followingthat the clutter texture is Gamma distributed. Since the textureand the speckle are independent, the covariance matrix of thedata ck is simply given by Σc = E(τk)Σx. For identifiability,we assume that E(τk) = 1. In the following, we investigatethe case where the data are either independent or follow aVARMA(0,1) model.

A. Independent observations

In this case, yk = ck =√τkxk which gives M = Σc. The

conditional PDF of yk conditionally to τk is

p(yk|τk) =1

πm|M |τmkexp

(−yH

k M−1yk

τk

). (12)

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

MFP

MCG

CML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

MFP

MCG

CML

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

MFP

MCG

CML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

MFP

MCG

CML

(a) (b) (c) (d)

Fig. 3. Pfa−λ plot and the corresponding Pd-SNR relationship for Pfa = 10−2 for spatially non correlated compound Gaussian clutter (ν = 0.5, m = 16,ρ = 0.5). (a), (b) K = 32. (c), (d) K = 48.

The value of τk that maximizes p(yk|τk) is τk =cHk Σ−1c ck/m = yH

k M−1yk/m, see e.g. [21]. Replacing τkby τk in (12), we get

L4(Y ;M) =

K∏k=1

mm exp(−m)

πm|M |(yHk M−1yk)m

. (13)

The partial derivative of (13) with respect to M leads to theTyler’s estimator [22] which is defined as the unique solutionof the following fixed point equation

MFP =m

K

K∑k=1

ykyHk

yHk M

−1FPyk

, (14)

and it can be noticed that this estimate does not depend onthe texture.

B. VARMA(0,1) observations

We assume that yk satisfies (4) where ck is compoundGaussian, and we investigate the conditional MLE’s of Θ1

and Σc obtained by assuming that c0 = 0. The conditionalPDF of ck conditionally to τk, p(ck|τk) is given by (12) whereyk is replaced by ck. Replacing τk by τk = cHk Σ−1c ck/m inp(ck|τk) and using the same arguments as in Section III-B1,we obtain the conditional likelihood function of Y ,

L5(Y ; Θ1,Σc) =

K∏k=1

mm exp(−m)

πm|Σc|(cHk Σ−1c ck)m. (15)

Since C = TY where T is given by (8) and involves only Θ1,the maximum of (15) with respect to Σc is Σc,FP given by(14) where MFP is replaced by Σc,FP. Then, replacing Σc byΣc,FP in (15) and maximizing with respect to Θ1, we obtainΘ

CG

1,CML. Replacing respectively Θ1 and Σc by ΘCG

1,CML and

Σc,FP in (5), we obtain MCG

CML.

V. RESULTS AND DISCUSSIONS

We have evaluated here both the conditional/exact likelihoodfunctions in Gaussian and in non-Gaussian cases for simulatedindependent or correlated data. All the results are given withm = 16 and K = 32 and 48 respectively and they arepresented with n = 104 Monte Carlo measurements. In thecase of Gaussian assumption, we maximize the likelihoodfunctions L1, L2 and L3 with respect to Σc and Θ1 simultane-ously. Whereas in the case of non Gaussian estimations, only

the maximization with respect to Θ1 is performed since theoptimal covariance matrix Σc is reached with the fixed pointalgorithm. For the exact likelihood function, the optimizationis initialized with initial values Θ1,0 and Σc,0 obtained fromthe Yule Walker estimator after transforming the VARMA(0,1)model to a Auto Regressive (AR) one [17]. As previously dis-cussed in section III in order to reduce calculation complexity,we assume here that the matrix Θ1 is a diagonal one and thatthe innovation covariance matrix Σc has a simple Toeplitzstructure (Σc)i,j = ρ|i−j| for i, j ∈ [1,m] with ρ = 0.5.

A. Estimation in Gaussian environment

The performance of ANMF (3) is evaluated in spatiallyuncorrelated and correlated Gaussian environments for fourcovariance matrix estimates: the benchmark (M is known),the SCM MSCM, the VARMA(0,1) conditional likelihoodestimate MCML and the VARMA(0,1) exact one MEML.For each detector, we plot the relationship between Pfa andthe detection threshold λ. Defining the Signal to Noise Ratio(SNR) as α2pHM−1p, these detectors are evaluated in termsof Pd-SNR performance for Pfa = 10−2. Figure 1 shows thecorresponding results for K = 32 and K = 48. In this figurewe can notice that the exact likelihood function calculated forVARMA(0,1) data gives best detection performance (exceptfor benchmark). This is explained by the fact that since thecovariance matrix Σc of the innovations has a specific Toeplitzstructure, we maximize the exact ML function with respect tothe factor ρ (only one parameter to estimate). This choice allowto speed-up the maximization task due to the simple Toeplitzstructure of the estimated Σc. The performance related to theconditional likelihood tends here to have the same performanceas SCM. The figure 2 shows the case of spatially correlatedVARMA(0,1) data for Θ1 = 0.9Im. In this figure, theconditional likelihood detector has better performance than theSCM one. This is explained by the fact that the SCM estimatordoes not take into account the spatial correlation of the clutter.Again, we can notice that the exact likelihood estimator givesthe nearest results to the optimal benchmarks obtained by thetheoretical covariance matrix M but this is always due tothe optimization constraint we have imposed to the particularToeplitz matrix.

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

MFP

MCG

CML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

MFP

MCG

CML

0 0.2 0.4 0.6 0.8 1

λ

10-4

10-3

10-2

10-1

100

Pfa

M

MFP

MCG

CML

-10 0 10 20 30

SNR (dB)

0

0.2

0.4

0.6

0.8

1

Pd

M

MFP

MCG

CML

(a) (b) (c) (d)

Fig. 4. Pfa−λ plot and the corresponding Pd-SNR relationship for Pfa = 10−2 for VARMA(0,1) spatially correlated compound Gaussian clutter (ν = 0.5,m = 16, ρ = 0.5, Θ1 = 0.9Im). (a), (b) K = 32. (c), (d) K = 48.

B. Estimation in compound Gaussian environment

The performance of ANMF (3) is evaluated in spatiallyuncorrelated and correlated compound Gaussian data environ-ment. Instead of the SCM estimate, we use here the Tyler’sestimator of the innovations covariance matrix as given in(14). The figure 3 shows the Pfa and the Pd values for bothTyler’s and conditional ML estimate of the covariance matrixfor K = 32 and K = 48. We can observe that the Tyler’sestimator is a little bit better than the conditional ML onefor small number of observations (K = 32). The seconddataset have been generated under highly spatially correlatedVARMA(0,1) model with Θ1 = 0.9Im. As in the case ofthe Gaussian data, the spatial correlation characterizing theclutter disrupts the accuracy of the Tyler’s estimator and theconditional ML gives slight better performance as illustratedin figure 4.

VI. CONCLUSION

In this paper, we proved that the spatial correlation char-acterizing the clutter secondary data could be exploited toenhance the performance of detection. This enhancement takesplace in simulated Gaussian distributed data through a max-imization of both conditional and exact likelihood functionsto estimate the data covariance matrix. In the case of nonGaussian distributed clutter, the conditional ML gives a con-siderable improvement of the quality of detection with respectto the classical Tyler’s estimator. The success of this estimatorwill motivate us to test it on real radar datasets and to estimatethe covariance matrix by calculating the non-Gaussian exactlikelihood function.

REFERENCES

[1] T. S. Rao and A. M. C. Antunes, “Spatio-temporal modelling oftemperature time series: a comparative study,” in Time series analysisand applications to geophysical systems, pp. 123–150, Springer, 2004.

[2] A. Cliff and J. K. Ord, “Space-time modelling with an application to re-gional forecasting,” Transactions of the Institute of British Geographers,pp. 119–128, 1975.

[3] M. Greco, F. Gini, and M. Rangaswamy, “Statistical analysis of mea-sured polarimetric clutter data at different range resolutions,” IEEProceedings-Radar, Sonar and Navigation, vol. 153, no. 6, pp. 473–481, 2006.

[4] A. L. Swindlehurst and P. Stoica, “Maximum likelihood methods inradar array signal processing,” Proceedings of the IEEE, vol. 86, no. 2,pp. 421–441, 1998.

[5] J. R. Roman, M. Rangaswamy, D. W. Davis, Q. Zhang, B. Himed, andJ. H. Michels, “Parametric adaptive matched filter for airborne radarapplications,” IEEE Transactions on Aerospace and Electronic Systems,vol. 36, no. 2, pp. 677–692, 2000.

[6] K. J. Sohn, H. Li, and B. Himed, “Parametric glrt for multichanneladaptive signal detection,” in Sensor Array and Multichannel Processing,2006. Fourth IEEE Workshop on, pp. 399–403, IEEE, 2006.

[7] E. Ollila, D. E. Tyler, V. Koivunen, and H. V. Poor, “Complex ellipticallysymmetric distributions: Survey, new results and applications,” SignalProcessing, IEEE Transactions on, vol. 60, pp. 5597 –5625, nov. 2012.

[8] P. J. Brockwell and R. A. Davis, Time series: theory and methods.Springer Science & Business Media, 2013.

[9] M. Deistler, W. Dunsmuir, and E. Hannan, “Vector linear time seriesmodels: corrections and extensions,” Advances in Applied Probability,vol. 10, no. 02, pp. 360–372, 1978.

[10] W. Dunsmuir, “A central limit theorem for parameter estimation instationary vector time series and its application to models for a signalobserved with noise,” The Annals of Statistics, pp. 490–506, 1979.

[11] G. C. Reinsel, Elements of multivariate time series analysis. SpringerScience & Business Media, 2003.

[12] E. J. Kelly, “An adaptive detection algorithm,” IEEE transactions onaerospace and electronic systems, no. 2, pp. 115–127, 1986.

[13] S. S. Haykin and A. O. Steinhardt, Adaptive radar detection andestimation, vol. 11. Wiley-Interscience, 1992.

[14] A. Aubry, A. De Maio, G. Foglia, C. Hao, and D. Orlando, “Radar detec-tion and range estimation using oversampled data,” IEEE Transactionson Aerospace and Electronic Systems, vol. 51, no. 2, pp. 1039–1052,2015.

[15] C. Hao, D. Orlando, G. Foglia, X. Ma, and C. Hou, “Adaptive radardetection and range estimation with oversampled data for partiallyhomogeneous environment,” IEEE Signal Processing Letters, vol. 22,no. 9, pp. 1359–1363, 2015.

[16] R. Couillet, M. S. Greco, J.-P. Ovarlez, and F. Pascal, “Rmt for whiteningspace correlation and applications to radar detection,” in ComputationalAdvances in Multi-Sensor Adaptive Processing (CAMSAP), 2015 IEEE6th International Workshop on, pp. 149–152, IEEE, 2015.

[17] R. S. Tsay, Multivariate Time Series Analysis: with R and financialapplications. John Wiley & Sons, 2013.

[18] J.-P. Ovarlez, F. Pascal, and P. Forster, “Covariance matrix estimation insirv and elliptical processes and their applications in radar detection,”2015.

[19] S. Kraut and L. L. Scharf, “The cfar adaptive subspace detector is ascale-invariant glrt,” IEEE Transactions on Signal Processing, vol. 47,no. 9, pp. 2538–2541, 1999.

[20] M. Rangaswamy, D. D. Weiner, and A. Ozturk, “Non-gaussian randomvector identification using spherically invariant random processes,” IEEETransactions on Aerospace and Electronic Systems, vol. 29, no. 1,pp. 111–124, 1993.

[21] F. Gini and M. Greco, “Covariance matrix estimation for CFAR detectionin correlated heavy tailed clutter,” Signal Processing, vol. 82, no. 12,pp. 1847–1859, 2002.

[22] D. Tyler, “A distribution-free M -estimator of multivariate scatter,” TheAnnals of Statistics, vol. 15, no. 1, pp. 234–251, 1987.