ON A SIRV-CFAR DETECTOR WITH RADAR EXPERIMENTATIONS IN IMPULSIVE CLUTTER

Transcription

1 O A SIRV-CFAR DETECTOR WITH RADAR EXPERIMETATIOS I IMPULSIVE CLUTTER Frédéric Pascal 23, Jean-Philippe Ovarlez, Philippe Forster 2, and Pascal Larzabal 3 OERA DEMR/TSI Chein de la Hunière, F-976, Palaiseau Cedex, France 2 Université Paris X - GEA Chein Desvallières, F-9240 Ville d Avray, France 3 ES Cachan - SATIE 6 Avenue du Président Wilson, F Cachan Cedex, France Eail : pascal@onera.fr, ovarlez@onera.fr, philippe.forster@cva.u-paris0.fr, pascal.larzabal@satie.ens-cachan.fr ABSTRACT This paper deals with radar detection in ipulsive clutter. Its ai is twofold. Firstly, assuing a Spherically Invariant Rando Vectors SIRV) odel for the clutter, the corresponding unknown covariance atrix is estiated by a recently introduced algorith [, 2]. A statistical analysis bias, consistency, asyptotic distribution) of this estiate will be suarized allowing us to give the GLRT properties: the SIRV-CFAR Constant False Alar Rate) property, i.e. texture-cfar and Covariance Matrix-CFAR, and the relationship between the Probability of False Alar PFA) and the detection threshold. Secondly, one of the ain contributions of this paper is to give soe results obtained with real non-gaussian data. These results deonstrate the interest of the proposed detection schee, and show an excellent correspondence between experiental and theoretical false alar rates.. PROBLEM STATEMET AD BACKGROUD on-gaussian noise characterization has gained any interests since experiental radar clutter easureents, ade by organizations like MIT [3], showed that these data are correctly described by non-gaussian statistical odels. One of the ost tractable and elegant non-gaussian odel coes fro the so-called Spherically Invariant Rando Vector SIRV) theory. A SIRV is the product of a Gaussian rando process - called speckle - with the square root of a non-negative rando variable - called texture. This odel leads to any results [4, 5]. The basic proble of detecting a coplex signal corrupted by an additive SIRV clutter c in a -diensional coplex vector y can be stated as the following binary hypothesis test: { H0 : y = c y i = c i i =,..., ) H : y = s+c y i = c i i =,..., where the c i s are signal-free independent easureents, traditionally called the secondary data, used to estiate the clutter covariance atrix. Under hypothesis H, it is assued that the observed data consists in the su of a signal s = α p and clutter c, where p is a perfectly known coplex steering vector and α is the signal coplex aplitude. Let us now recap soe SIRV theory results. A noise odelled as a SIRV is a non-hoogeneous Gaussian process with rando power. More precisely, a SIRV [6] is the product of the square root of a positive rando variable τ texture) and a -diensional independent coplex Gaussian vector x speckle) with zero ean covariance atrix M = Exx H ) with noralization TrM) =, where H denotes the conjugate transpose operator where The SIRV PDF expression is g c,τ) = c = τ x. 2) + p c) = g c,τ) pτ)dτ, 3) 0 π τ) M exp ch M ) c. 4) τ This odel allowed to build several Generalized Likelihood Ratio Tests like the GLRT-Linear Quadratic GLRT- LQ) in [4, 5] defined by p H M y 2 H ΛM) = p H M p)y H M λ, 5) where p is the steering vector, y the observed vector and λ the detection threshold associated to this detector. In any probles, non-gaussian noise can be characterized by SIRVs but the covariance atrix M is generally not known and an estiate M is required. Obviously, it has to satisfy the M-noralization: Tr M) =. The next section is devoted to an adaptive GLRT built fro an Approxiate Maxiu Likelihood AML) estiate of the SIRV covariance atrix. Then, Section 3 presents an application of this detector to real data: experiental results perfectly atch theoretical analysis.

2 2. THE FIXED POIT ESTIMATE M FP AD THE CORRESPODIG ADAPTIVE GLRT 2. The AML estiate Conte and Gini in [, 2] have shown that an Approxiate Maxiu Likelihood AML) estiate M of M is a solution of the following equation M = ci c H ) i i= M. 6) c i otice that the ML estiate has been studied in [7]. Existence and uniqueness of the above equation solution, denoted M FP have already been investigated in [8]. c H i Let the function f be defined as f M) = i= ci c H i c H i M c i ) = i= xi x H i x H i M x i ), 7) where the right hand side of 7) is rewritten in ters of the x i s and the τ i s. Eqn. 7) obviously iplies that M FP is independent of the τ i s. The statistical properties of M FP have been investigated and published in [], the ain results are recaped here below: Proposition 2. ) MFP is a consistent estiate of M; 2) M FP is an unbiased estiate of M; 3) the asyptotic distribution of M FP is Gaussian and its covariance atrix is fully characterized in []; 4) this distribution is the sae as the) asyptotic distribution of a Wishart atrix with degrees of freedo The studied adaptive GLRT Let us now present the adaptive GLRT [9, 0], used for detection p ˆΛ M) H = M y 2 H p H M p)y H M λ. 8) In the next section dealing with applications to real data, we will use ˆΛ M FP ) as detector. This detector is obviously texture-cfar independent of the distribution of τ) and, an original result of this paper is to show the independence of the distribution of ˆΛ M FP ) with the covariance atrix M: we will say that ˆΛ M FP ) is Matrix-CFAR M-CFAR). Definition 2. An adaptive detector ˆΛ M) verifies the M-CFAR property if its statistical distribution is independent of the covariance atrix M estiated by M. This property is of ost interest in a practical work to detect targets when the covariance atrix is unknown. Theore 2. ˆΛ M FP ) is M-CFAR. Theore 2. establishes the M-CFAR property of the adaptive GLRT built with the FP estiate. Proof 2. Let M be a covariance atrix. Let M FP be the FP estiate of M. Then, under hypothesis H 0 no target signal), we will show that L ˆΛ M FP ) ) ) = L ˆΛ M FP,I ) 9) where L x) stands for the statistical distribution of a rando variable x and M FP,I is the FP estiate of the identity atrix I. Since the statistics of ˆΛ M FP ) is independent of the texture τ, we choose τ = : secondary data x,...,x are thus Gaussian with covariance atrix M, x i 0,M). The FP estiate of M is defined as the unique solution up to a scalar factor) of M FP = i= x H i and the adaptive GLRT detector is ˆΛ M FP ) = p x i x H i p H M H M FP M, 0) FP x i FPx 2 p)xh M FP x) H H 0 λ, where x is the observation vector under hypothesis H 0 ) such that x 0,M). The first part of the proof is the whitening of the data. By applying the following change of variable, y = M /2 x to Eqn. 0), one has where Therefore, M FP = i= M /2 y i y H i M /2 y H i T = M /2 MFP M /2. T = i= y H i T y i, ) y i y H i T. 2) y i T is thus the unique FP estiate up to a scalar factor) of the identity atrix. Its statistics is clearly independent of M since the y i s are 0,I). Moreover, for any unitary atrix U, one has

3 U TU H = i= z H i z i z H i U TU H), 3) zi 0 Clutter Map where z i = Uy i is also 0,I). Therefore, U TU H has the sae distribution as T. In ters of the adaptive detector, one has p ˆΛ M H FP ) = T y 2 H p H T p )y H T λ, where p = M /2 p and y = M /2 x is 0,I). ow let U be a unitary atrix such that Up = p e 4) where e = ), denotes the transpose operator and p denotes the 2-nor of vector p. Aziuth Range bins Figure : Ground clutter data level in db) corresponding to the first pulse echo. Y-coordinates represent 70 aziuth angles and X-coordinates represent = 868 range bins. Thanks to Eqn. 4), one has ˆΛ M FP ) = e H U TU H) z 2 e H U TU H) e)z H U TU H) z) H H 0 λ, where z = Uy is also 0,I). By setting M FP,I = U TU H, we see that the distribution of ˆΛ M F P) does not depend on M, which copletes the proof of Theore 2.. ote also that the distribution of ˆΛ M F P) does not depend on the steering vector p. In this section, the statistical perforance of the FP estiate has been investigated as well as the SIRV-CFAR texture-cfar and M-CFAR) property of the adaptive GLRT, built with M FP. One of the first deduction of previous results is that whatever the SIRV used, for different distributions of the texture and for different covariance atrices, the resulting GLRT ˆΛ M) follows the sae distribution. This is of a ajor interest in areas of clutter transition like for exaple, in coastal areas ground and sea) or at the edge of forests fields and trees) because the detector should be insensitive to the different clutter areas. This is the object of the next section. 3. RADAR APPLICATIOS This section is devoted to the analysis of different radar easureents in which the clutter is strongly ipulsive. In a first tie, let us give soe generalities. In radar detection, the analysis falls into two independent stages: The calculation of the detection threshold λ to ensure a false alar rate, given by the operator. This part is perfored by a learning of the clutter. The coparison of the adaptive GLRT ˆΛ M) with the detection threshold. Let us define soe notations: the Probability of False Alar the Probability of Detection PD) P f a = PΛ > λ H 0 ), 5) P d = PΛ > λ H ). 6) In [2], a theoretical relationship between the detection threshold λ and the PFA has been established when the covariance atrix M is estiated by the well known Saple Covariance Matrix SCM) estiate defined by M SCM = i= c i c H i. 7) ow, the expression of PFA-threshold relationship in this specific case SCM estiate) is P f a = λ) a 2F a,a ;b ;λ), 8) where a = +2, b = + 2 and 2 F is the hypergeoetric function [3] defined as 2F a,b;c;x) = Γc) Γa)Γb) k=0 Γa+k)Γb+k) x k Γc+k) k!. 9) Moreover, thanks to point 4) of proposition 2. and since the SCM 7) is Wishart distributed [4], expression 8 still holds for large, when the covariance atrix M is estiated by the FP estiate: P f a = λ) a 2F a,a ;b ;λ), 20) where a = +2 and b = + 2. It eans + + that it is the sae relationship but with less secondary data data instead of in the Wishart case). +

4 tection threshold λ for a given PFA. A coon procedure is to set this threshold, which is a syste design paraeter, based on the designer perception of tradeoffs between false alars and issed detection. Traditionally, the experiental detection threshold adjustent is deterined by counting, by oving a rectangular CFAR-ask of size 5 5. For all central cells of the ask i.e. the cell under test), the dark cell on Figure 3, corresponding to the studied observation y 8- vector), a value of ˆΛ M) is calculated. The covariance atrix M has been estiated with the set of = 24 8-vectors, considered as the secondary data, y,...,y 24, and situated around the tested cell. These reference cells are the light blue cells on Figure 3. Figure 2: Ground clutter data level in db) corresponding to the first pulse echo. Y-coordinates represent 70 aziuth angles and X-coordinates represent = 868 range bins. This result has never been validated on real easureents: this is one of the purposes of this section. The ground clutter data presented in this paper were collected by an operational radar at THALES Air Defence, placed 3 eters above ground and illuinating the ground at low grazing angle. Ground clutter coplex echoes were collected in = 868 range bins for 70 different aziuth angles and for = 8 recurrences, which eans that vectors size is = 8. ear the radar, echoes characterize non-gaussian heterogeneous ground clutter whereas beyond the radioelectric horizon of the radar around 5 ks) only Gaussian theral noise the dark part of the ap) is presented Figure ). To ephasize the areas of ipulsive clutter, Figure 2 represents in 3 diensions, the sae range bin-aziuth ap as on Figure : the third diension vertical) shows the power of the clutter. 3. Validation of Eqn.20) on real data One purpose of this paper is to validate the theoretical relationship 20) between the detection threshold and the PFA thanks to counting of the real data when the covariance atrix is estiated by the FP estiate. Moreover, when it is assued that the covariance atrix M is known, one has λ = P f a see for exaple [5]). otice that this equation has just a theoretical interest because in practice, M is always unknown. Reark 3. ote that the counting syste on the real data akes sense only thanks to the M-CFAR property of the adaptive detector. Indeed, there is no valid reason why all the sets of 24 data have the sae covariance atrix Curves "PFA threshold" Fixed Point M hat M known PFA Detection threshold λ Figure 4: Validation of PFA-threshold relationship Figure 3: CFAR ask The analysis of these radar data allows to adjust the de- Authors are grateful to Thales Air Defence for the analysis of their data On Figure 4, the solid curve corresponds to the theoretical relationship PFA-threshold if M is known while the dotted curve represents the theoretical relationship PFAthreshold when M and is assued unknown and estiated by M FP. The curve ade of crosses ) represents the experiental ade with CFAR asks by counting) relationship PFA-threshold when M is estiated by M FP. It perfectly atches the theoretical relationship. Obtaining this result has

5 been possible only because the detector ˆΛ M FP ) satisfies the M-CFAR property, essential in an heterogeneous clutter. Thus, this application validates Eqn. 20) and an essential consequence of this result is that thanks to Eqn. 20), the clutter training is not essential any ore for the adjustent of the detection threshold. 3.2 Robustness to the clutter transitions Figure 5 presents, for all the points of the range bin-aziuth ap, the GLRT calculated for the FP estiate M FP. This ap was ade fro the 8 available aps of real data. Figure 5: Detector ap We can conclude fro Figure 5 that in spite of the clutter heterogeneity, on the right hand side of Figure, the use of the FP estiate in the GLRT allows to obtain a copletely unifor likelihood ratio ap. This experiental result ensures a constant false alar regulation, even in the transitions areas. Moreover, it is in a good agreeent with the theory and is directly provided by the SIRV-CFAR property of ˆΛ M FP ). 4. COCLUSIO In this paper, the M-CFAR property of the adaptive detector GLRT ˆΛ M FP ) built with the FP estiate of the covariance atrix M has been established. This result has been used in a radar application on real non-gaussian data. This property stated the independence between the GLRT distribution and the real covariance atrix M of the data. Moreover, a goal of this paper was the analysis of non- Gaussian experiental ground clutter signals. For that purpose, we first validated a theoretical relationship between the detection threshold and the PFA established in [2], thanks to non-gaussian data. Then, we highlighted the GLRT robustness to the clutter transitions thanks to it SIRV-CFAR property. Acknowledgeent The authors are grateful to Thales Air Defence for the analysis of their experiental data. REFERECES [] F. Gini and M. V Greco, Covariance atrix estiation for CFAR detection in correlated heavy tailed clutter, Signal Processing, special section on Signal Processing with Heavy Tailed Distributions, vol. 82, no. 2, pp , Dec [2] E. Conte, A. De Maio and G. Ricci, Recursive estiation of the covariance atrix of a copound-gaussian process and its application to adaptive CFAR detection, IEEE Trans. Signal Process., vol. 50, no. 8, pp , Aug [3] J.B. Billingsley, Ground Clutter Measureents for Surface-Sited Radar, Technical Report 780, MIT, February 993. [4] E. Conte, M. Lops and G. Ricci, Asyptotically Optiu Radar Detection in Copound-Gaussian Clutter, IEEE Trans.-AES, vol. 3, no. 2, pp , April 995. [5] F. Gini, Sub-Optiu Coherent Radar Detection in a Mixture of K-Distributed and Gaussian Clutter, IEE Proc.Radar, Sonar avig, vol. 44, no., pp , February 997. [6] K. Yao A Representation Theore and its Applications to Spherically Invariant Rando Processes, Trans.-IT, vol. 9, no. 5, pp , Septeber 973. [7] M. Rangasway Statistical analysis of the nonhoogeneity detector for non-gaussian interference backgrounds, Trans.-SP, vol. 53, no. 6, pp. 20 2, June [8] F. Pascal, Y. Chitour, J.P. Ovarlez, P. Forster and P. Larzabal Covariance Structure Maxiu Likelihood Estiates in Copound Gaussian oise : Existence and Algorith Analysis, under review at IEEE Trans.-SP. [9] E. Conte, M. Lops, G. Ricci, Adaptive Radar Detection in Copound-Gaussian Clutter, Proc. of the European Signal processing Conf., Edinburgh, Scotland, Septeber 994. [0] F. Gini, M.V. Greco, and L. Verrazzani, Detection Proble in Mixed Clutter Environent as a Gaussian Proble by Adaptive Pre-Processing, Electronics Letters, vol. 3, no. 4, pp , July 995. [] F. Pascal, P. Forster, J.P. Ovarlez and P. Larzabal, Theoretical analysis of an iproved covariance atrix estiator in non-gaussian oise, IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, PA, USA, March [2] F. Pascal, J.P. Ovarlez, P. Forster and P. Larzabal, Constant False Alar Rate Detection in Spherically Invariant Rando Processes, Proc. of the European Signal processing Conf., Vienna, Austria, Septeber 2004, pp [3] M. Abraowitz and I.A. Stegun, Handbook of Matheatical Functions, ational Bureau of Standard, AMS 55, June 964. [4] A. K. Gupta and D. K. agar, Matrix Variate Distributions, Chapan & Hall/CRC, Monographs and Surveys in Pure and Applied Matheatics, 04, August 999. [5] E. Jay, J.P. Ovarlez, D. Declercq and P. Duvaut, BORD : Bayesian Optiu Radar Detector, Signal Processing, vol. 83, no. 6, pp. 5 62, June 2003.