OPEN ACCESS
In the STAP framework, this paper proposes a review of SIRV modelling for detection and estimation. These processes allow to extend the detection and estimation theory to the STAP problem in a non-Gaussian and heterogeneous background. Some theoretical tools (estimators, detectors) are presented jointly with their attractive properties (SIRV CFAR, robustness, secondary data number reduction) which allow to significantly outperform conventional, e.g. Gaussian based, ones.
Extended Abstract
Space Time Adaptive Processing (STAP) is a recent technique used in airborne phased array radar to detect moving targets embedded in an interference background such as jamming or strong clutter. While conventional radars are capable of detecting targets both in the time domain related to target range and in the frequency domain related to target velocity, STAP uses an additional domain (space or information collected by an antennas array) related to the target angular localization. The joint processing of these space-time data, by appropriate two-dimensional adaptive filtering methods, allows stronger interference/clutter rejection and therefore improved target detection. Moreover, STAP can now be used in High Resolution (STAP-HR) for a better classification of the moving target. But in this case, the widely used hypothesis of a Gaussian clutter is not valid anymore because of the spatial heterogeneity of the clutter (clutter power, called the texture, may change spatially) and detection performance significantly decreases. For example, in the area of STAP High Resolution (STAP-HR), the resolution is such that the Central Limit Theorem cannot be applied anymore since the number of backscatters is too small. Equivalently, it is known that reflected signals could be very impulsive in low grazing angle radar. This is why, in the last decades, the radar community has been very interested on problems dealing with non-Gaussian clutter modeling.
To fill these gaps, non-Gaussian models for the clutter have to be considered. In the literature of radar detection and estimation, the Spherically Invariant Random Vector (SIRV) modelling is generally used for its good statistical properties and for its good fitting to real data. More precisely, a SIRV is the product of a Gaussian random process with the square root of a non-negative random scalar variable (the so-called texture). Thus, the SIRV is fully characterized by the texture (representing an unknown power) and the unknown covariance matrix of the zero-mean Gaussian vector. One of the major challenging difficulties in SIRV modeling is to estimate these two unknown quantities and particularly the Clutter Covariance Matrix for STAP applications of our concern, better the accuracy of this estimate, better the detection performance. Furthermore, this model provides a very general framework which includes classical distributions as for example the Gaussian distribution, the K-distribution or the Weibull distribution. The Clutter Covariance Matrix is estimated from signal-free and independent data, called the secondary data. Under nonGaussian clutter assumptions, this involves several difficulties. First, the spatial heterogeneity of the clutter deteriorates the Clutter Covariance Matrix estimation accuracy. Moreover, if the secondary data are not samples of the same parameterized distribution, i.e. the same covariance matrix, this estimation makes no sense: this is the case for example in non-side looking configuration where the Clutter Covariance Matrix is non stationary from cell to cell: this drawback is called frequency heterogeneity clutter. Secondary data may also be corrupted by a high number of targets, interferences or jamming. The conventional techniques based on the empirical estimation of the Clutter Covariance Matrix are therefore not robust.
We propose in this paper to analyze and to solve, in the SIRV theory framework and more generally in the context of robust statistics, these typical effects (spatial heterogeneity of the clutter intensity, clutter heterogeneity, influence of the number of secondary data, influence of the contaminated secondary date) arising in modern STAP analysis.
In the first part of the paper, we propose to describe the SIRV modelling for clutter background. In this context, all the conventional (Gaussian background) and more recent techniques on detection and parameters estimation in non-Gaussian background are recalled. This leads to define a new kind of detector (Adaptive Normalized Matched Filter) and a new kind of Clutter Covariance Matrix estimation (Fixed Point Covariance Matrix). Jointly combined, they characterize a new STAP detector which has the SIRV-CFAR property. This latter property means that this detector under clutter assumption only does not depend on the texture and on the covariance matrix. It can so be robust when dealing with clutter heterogeneity, clutter transition and so warrants Constant False Alarm Rate. The second part describes the robustness of the proposed detector with respect to targets, jammers, strong echoes present in the contaminated secondary data. The third part will focus on the a priori hypothesis that can be made on the structure of the Clutter Covariance Matrix, i.e., the persymmetry property that characterizes the STAP in side-looking configuration. This property will lead to reduce by a factor two the number of secondary data needed to estimate the covariance matrix and this for the same performance. The latter detector, taking into account this property of persymmetry will so be extended in SIRV context. The last part is devoted to the STAP results obtained on experimental data provided by the DGA/MI. They show that the proposed STAP detector has the same performance in a classical Gaussian background but can significantly improves them in non-Gaussian clutter.
All the presented works are here focused on the STAP application. However, since the theoretical tools are very general, they can be easily extended for other radar applications (SAR images classification, MIMO radar, Polarimetry, hyperspectral data, etc.).
RÉSUMÉ
Ce papier décrit l’intérêt de l’apport des processus SIRV (processus gaussiens composés) pour la détection et l’estimation dans le cadre de la détection de cibles mobiles (STAP). Ces processus permettent de manière élégante et efficace d’étendre au problème STAP la théorie de la détection et de l’estimation dans un cadre de bruit additif (fouillis) hétérogène et/ou non gaussien. Nous présentons ainsi les outils théoriques (estimateurs, détecteurs) qui permettent d’améliorer les performances des détecteurs conventionnels ainsi que leurs propriétés (SIRV TFAC, robustesse, réduction du nombre de données secondaires).
adaptive signal detection, STAP, Gaussian and non-Gaussian clutter, SIRV theory, covariance matrix estimation, persymmetry, robust estimation
MOTS-CLÉS
détection adaptative, STAP, fouillis gaussien et non-gaussien, théorie des SIRV, estimation de matrice de covariance, persymétrie, estimation robuste
Abramowitz, M., Stegun, I., C., (1972). Handbook of mathematical functions, New York, Dover.
Barnard T.J., Weiner D.D., (1996). Non-Gaussian clutter modeling with generalized spherically invariant random vectors, IEEE Trans. on Signal Processing, vol. 44, n° 10, p. 2384-2390.
Bausson S., Pascal F., Forster P., Ovarlez J.P., Larzabal P., (2007). First and second order moments of the normalized sample covariance matrix of spherically invariant random vectors, IEEE Signal Processing Letters, vol. 14, n° 6, p. 425-428.
Bidon S. (2011). Introduction au STAP. Partie II : Modèle des signaux et principe du filtrage, Revue Traitement du Signal, vol. 28, n° 1-2, p. 35-56.
Bidon S., Montécot M., Savy L., (2011). Introduction au STAP. Partie III : Les données du club STAP, Revue Traitement du Signal, vol. 28, n° 1-2, p. 57-79.
Bidon S., Besson O., Tourneret J.Y., (2011). Synthèse des traitements STAP pour la détection en environnement hétérogène, Revue Traitement du Signal, vol. 28, n° 1-2, p. 81-112.
Billingsley J.B., (1993). Ground clutter measurements for surface-sited radar, Technical Report 780, MIT, février 1993.
Billingsley J.B., (2002). Low-Angle Radar Land Clutter. Measurements and Empirical Models, Norwitch, NY, William Andrew Publishing.
Bilodeau M., Brenner D., (1999). Theory of multivariate statistics, Springer Verlag.
Brennan L.E., Reed I.S., (1973). Theory of Adaptive Radar, IEEE Trans. on Aerospace and Electronic Systems, vol. 9, n° 2, p. 237-252.
Burg J.P., Luenberger D.G., Wenger D.L., (1982). Estimation of Structured Covariance Matrices, Proc. of the IEEE, vol. 70, p. 963-974.
Cai L., Wang H., (1992). A persymmetric multiband GLR algorithm, IEEE Trans. on Aerospace and Electronic Systems, vol. 28, n° 3, p. 806-816.
Conte E., Lops M., Ricci G., (1995). Asymptotically Optimum Radar Detection in CompoundGaussian Clutter, IEEE Trans. on Aerospace and Electronic Systems, vol. 31, n° 2, p. 617-625.
Conte E., Ricci G., (1998). Sensitivity study of GLRT detection in compound-Gaussian clutter, IEEE Trans. on Aerospace and Electronic Systems, vol. 34, n° 1, p. 308-316.
Conte E., Lops M., Ricci G., (1999). Incoherent radar detection in compound-Gaussian clutter, IEEE Trans. on Aerospace and Electronic Systems, vol. 35, n° 3, p. 790-800.
Conte E., De Maio A., Galdi, C., (2000). Signal detection in compound-Gaussian noise: Neyman-Pearson and CFAR detectors, IEEE Trans. on Signal Processing, vol. 48, n° 2, pp. 419-428.
Conte E., De Maio A., Ricci G., (2002). Recursive estimation of the covariance matrix of a compound-Gaussian process and its application to adaptive CFAR detection, IEEE Trans. on Signal Processing, vol. 50, n° 8, p. 1908-1915.
Conte E., De Maio A., (2003). Exploiting persymmetry for CFAR detection in compoundGaussian Clutter, IEEE Trans. on Aerospace and Electronic Systems, vol. 39, n° 2, p. 719-724.
Conte E., De Maio A., (2004). Mitigation techniques for non-Gaussian sea clutter, IEEE Journal of Oceanic Engineering, vol. 29, n° 2, p. 284-302.
Fuhrmann D.R., (1991). Application of Toeplitz covariance estimation to adaptive beamforming and detection, IEEE Trans. on Signal Processing, vol. 39, p. 2194-2198.
Gini F., (1997). Sub-optimum coherent radar detection in a mixture of K-distributed and Gaussian clutter, IEE Proc. Radar, Sonar and Navigation, vol. 144, n° 1, p. 39-48.
Gini F., Greco M.V., (2002). Covariance matrix estimation for CFAR detection in correlated heavy tailed clutter, Signal Processing, Special section on Signal Processing with Heavy Tailed Distributions, vol.. 82, n° 12, p. 1847-1859.
Ginolhac G., Forster P., Ovarlez J.-P., Pascal F., (2011). STAP à rang réduit, robuste et persymétrique, Revue Traitement du Signal, vol. 28, n° 1-2, p. 143-170.
Huber P.J., Ronchetti E. M., (2009), Robust statistics, John Wiley & Sons Inc.
Jakeman E., (1980). On the Statistics of K-Distributed Noise, J. Phys. A: Math Gen., vol. 13, p. 31-48.
Jao J.K., (1984). Amplitude distribution of composite terrain radar clutter and the K-distribution, IEEE Trans. on Antennas and Propagation, vol. 32, n° 10, p. 1049-1062.
Jay E., (2002). Détection en environnement non-gaussien, Thèse de doctorat. Université de Cergy Pontoise. Kelker, D., Distribution theory of spherical distributions and a location-scale parameter generalization, (1970). Sankhya A, vol. 32, pp. 419-430.
Kelly E.J., (1986). An adaptive detection algorithm, IEEE Trans. on Aerospace and Electronic Systems, vol. 23, pp.115-127.
Korado V.A.,(1968). Optimum detection of signals with random parameters against the background of noise of unknown intensity under conditions of constant false alarm probability, Radio Engineering and Electronic Physics, vol. 13, p. 671-679.
Kraut S., Scharf L.L., (1999). The CFAR adaptive subspace detector is a scale-invariant GLRT, IEEE Trans. on Signal Processing, vol. 47, n° 9, p. 2538-2541.
Kraut S., Scharf L.L., Mc Whorter L.T., (2001). Adaptive subspace detectors, IEEE Trans. on
Signal Processing, vol. 49, n° 1, p. 1-16.
Mahot M., Forster P., Ovarlez J.P., Pascal F., (2010). Robustness analysis of covariance matrix estimates, Actes EUSIPCO’10, Aalborg, Danemark, 23-27 août 2010.
Maronna R. A., (1976). Robust M-estimators of multivariate location and scatter, Annals of Statistics, vol. 4, n° 1, pp. 51-67.
Montécot M., Le Chevalier F., Savy L., (2011), Introduction au STAP. Partie I : Contexte radar et enjeu du filtrage, Revue Traitement du Signal.
Nitzberg R., Burke J.R., (1980). Application of maximum likelihood estimation of persymmetric covariance matrices to adaptive detection, IEEE Trans. on Aerospace and Electronic Systems, vol. 25, p. 124-127.
Pailloux G., (2010). Estimation structurée de la covariance du bruit en détection adaptative, Thèse de doctorat. Université de Paris 10.
Pailloux, G., Forster P., Ovarlez J.P., Pascal F., (2008). On persymmetric covariance matrices in adaptive detection, Actes IEEE-ICASSP’08, Las Vegas, USA, p. 2305-2308.
Pailloux G., Ovarlez J.P., Pascal F., Forster P., (2008). A SIRV-CFAR adaptive detector exploiting persymmetric clutter covariance structure, Actes IEEE Radar Conference, Rome, Italie, p. 1139-1144.
Pailloux G., Pascal F., Ovarlez J.P., Forster P., (2009). Comparison of various detection schemes for STAP radar based on experimental data, Actes IEEE-Radar’09, Bordeaux, France, 12-16 octobre 2009.
Pailloux G., Forster P., Ovarlez J.P., Pascal F., (2011). Persymmetric adaptive radar detectors, IEEE Trans. on Aerospace and Electronic Systems, à paraître.
Pascal F., Ovarlez J.P., Forster P. et Larzabal P., (2004). Constant false alarm rate detection in spherically invariant random processes, Actes EUSIPCO’04, Vienne, Autriche, 6-10 septembre 2004, p. 2143-2146.
Pascal F., Ovarlez J.P., Forster P. et Larzabal P., (2004). Radar detection In compoundGaussian clutter, Actes IEEE Radar’04, Toulouse, France, 18-22 octobre 2004.
Pascal F., Ovarlez J.P., Forster P. et Larzabal P., (2005). Theoretical analysis of an improved covariance matrix estimator in non-Gaussian noise, Actes IEEE-ICASSP’05, Philadelphie, USA, 18-23 mars 205.
Pascal F., (2006). Détection et estimation en environnement non-gaussien, Thèse de doctorat. Université de Paris 10.
Pascal F., Ovarlez J.P., Forster P., Larzabal P., (2006). On a SIRV-CFAR detector with radar experimentations in impulsive noise, Actes EUSIPCO’06, Florence, Italie, 4-8 septembre 2006.
Pascal F., Chitour Y., Ovarlez J.P., Forster P., Larzabal P., (2008). Covariance structure maximum likelihood estimates in compound Gaussian noise: existence and algorithm analysis, IEEE Trans-SP, vol. 56, n° 1, p. 34-48.
Pascal F., Forster P., Ovarlez J.P., Larzabal P., (2008). Performance analysis of covariance matrix estimates in an impulsive noise, IEEE Trans. on Signal Processing, vol. 56, n° 6, p. 2206-2217.
Picinbono B., Vezzosi G., (1970). Détection d’un signal certain dans un bruit non stationnaire et non-gaussien, Annales des Télécommunications, vol. 25, p. 433-439.
Rangaswamy M., Weiner D.D., Öztürk A., (1993). Non-Gaussian random vector identification using spherically invariant random processes”, IEEE Trans. on Aerospace and Electronic Systems, vol. 29, n° 1, p. 111-123.
Reed I.S., Mallet J.D., Brennan L.E, (1974). Rapid convergence rate in adaptive arrays, IEEE Trans. on Aerospace and Electronic Systems, vol. 10, n° 10, p .853-863.
Robert C.P., (1994). The Bayesian Choice: a Decision-Theoretic Motivation, New York, Springer-Verlag.
Robey F.C., Fuhrmann D.R., Kelly E.J., Nitzberg R., (1992). A CFAR adaptive matched filter detector, IEEE Trans. on Aerospace and Electronic Systems, vol. 28, n° 1, p. 208-216.
Sangston K.J., Gini F., Greco M.V., Farina A., (1999). Structures for radar detection in compound gaussian clutter, IEEE Trans. on Aerospace and Electronic Systems, vol. 35, n° 2, p. 445-458.
Scharf L.L., Friedlander B., (1994). Matched subspace detectors, IEEE Trans. on Signal Processing, vol. 42, n° 8, p. 2146-2157.
Scharf L.L., Lytle D.W., (1971). Signal detection in Gaussian noise of unknown level: an invariance application, IEEE Trans. on Information Theory, vol. 17, p. 404-411.
Tyler D., (1983). Robustness and efficiency properties of scatter matrices, Biometrika, vol. 70, n° 2, p. 411.
Vasile G., Ovarlez J.P., Pascal F., Tison C., (2010). Coherency matrix estimation of heterogeneous clutter in high resolution polarimetric SAR images, IEEE Trans. on Geoscience and Remote Sensing, vol. 48, n° 4, p. 1809-1826.
Yao K., (1973). A representation theorem and its applications to spherically invariant random aprocesses, IEEE Trans. on Information Theory, vol. 19, n° 5, p. 600-608.