OPEN ACCESS
In the STAP domain, modeling the interferences as an autoregressive (AR) process with the detector called Parametric Adaptive Matched Filter (PAMF), provides an estimation of the clutter-rejection filter with few training data. The main difficulty of this approach is the estimation of the AR matrices by using the training data. Thus, we propose an on-line estimation based on the Kalman filter and its variants. A comparative study is carried out and illustrates the relevance of such approaches with data provided by the DGA.
Extended Abstract
The purpose of Radar (RAdio Detection And Ranging) is mainly to detect, to
locate and to track targets. When coherent-pulse radar is used, the target velocity can be also estimated. For surveillance mission, airborne radars are required to provide long-range detection of increasingly smaller targets. Unfortunately, two main problems occur. Firstly, the long-range detection may be difficult due to the decay of the reflected power. Secondly, the jammer, the thermal noise and the clutter which is due to the atmosphere, the rain, the ground or the sea can disturb the return of the slow-moving target.
To address the above problems, adaptive radar, the theory of which was proposed in various papers by Brennan, Mallett and Reed in the 70’s, can be considered. More particularly, when a phased array antenna system is used, space-time adaptive processing (STAP) consists in simultaneously combining the signals which are received by multiple elements of an antenna array and which result from multiple pulse repetition periods of a coherent processing interval. More specifically, STAP aims at maximizing the signal-to-interference-plus-noise ratio (SINR). Then, the filter output is used to define a test statistic that makes it possible to detect whether there is a target or not in the cell under test.
For a few years, several variants have been proposed to both reduce the computational cost, the number of training data and to make the detection more robust against interfering targets. Among them, the so-called “Partially” STAP approaches take into account the characteristics of the received data. When dealing with the subspace techniques, a projection into the interference-free subspace can be done. Another approach consists in modeling the interferences as an multichannel autoregressive (AR) process used with a specific detector called Parametric Adaptive Matched Filter (PAMF). This algrithm provides an estimation of the clutter-rejection filter with few training data. This estimation is also robust against interfering targets and the non-stationarity of the clutter. Nevertheless, the main difficulty of this approach is probably the estimation of the AR matrices by using the training data.
Therefore, in this paper, we propose to study two cases:
Firstly, if the interferences are only composed of a Gaussian clutter, well-known off-line estimation approaches such as the multichannel Yule-Walker, the Maximum Likelihood (ML) estimation or the Nuttall-Strand method can be considered. As an alternative, Kalman filter can be used as it is the minimum mean square error estimator of the AR matrices.
Secondly, if the interferences are composed of thermal noise and Gaussian clutter, the clutter is modeled as a multichannel AR process disturbed by a thermal noise. In that case, we focus our attention on the most disturbing interference, namely the clutter. However, the parameter estimation issue may be a hard task. Off-line noisecompensating algorithms have been proposed by Mahmoudi, Zheng, etc. in other fields than radar processing. In that case, the variance of the additive noise and the parameters are jointly estimated. Nevertheless, Zheng’s method is no longer reliable when the signal-to-noise ratio (SNR) becomes lower than 10 dB. Indeed, it may lead to a set of AR parameter matrix estimates corresponding to an unstable system. Errors-in Variables approaches can be also considered and provide both the variancesof the thermal noise, the AR matrices and the covariance matrix of the driving process. However, the computational cost is high. For the above reasons, we propose to study the relevance of on-line approaches based on Kalman filtering in this paper.In that case, the state space representation of this system comprises two equations,the update equation of which is not linear. To estimate the parameters, we first suggest using the extended Kalman filter. It is based on the 1st order Taylor expansion of the state space representation of the system around the last state vector estimate.Then,we study the Sigma Point Kalman Filters, which include two methods:
1/ the unscented Kalman filter, which is based on the unscented transformation
initially proposed by Julier,
2/ the central difference Kalman filter, which is based on the Stirling interpolation formula to derive another expression for the mean and the covariance matrix estimations of the posterior variable.
Then, a comparative study is carried out and illustrates the relevance of such approaches with eleven sets of real data provided by the DGA. To evaluate the performances of the correlation method, the EKF and the UKF, the SINR is estimated at the output of the PAMF. The PAMF combined with the UKF estimation gives the highest SINR for seven sets of data. In addition, this method requires only four range cells as training data and is quite robust against the non Gaussian assumptions of clutter. In addition, with its square-root form allowing its computational cost to be lower than the standard UKF, its use is relevant for embedded systems.
REMARK. – It is true that other variants of Kalman filtering such as Iterative Kalman filtering, 2nd order extended Kalman filtering and quadrature Kalman filtering could be considered. To reduce the computational cost, the so-called Sigma rho Kalman filter could be used. To relax the Gaussian assumptions, H∞ filter and its variants in the non-linear case such as the extended H∞ filtering or an Unscented H∞ filtering could be studied.
RÉSUMÉ
Dans le cadre du traitement STAP, une modélisation autorégressive (AR) des
interférences utilisée avec un détecteur appelé Parametric Adaptive Matched Filter (PAMF) donne lieu à un filtre de réjection du fouillis pour lequel le domaine d’entraînement est réduit.La principale difficulté de cette approche réside alors dans l’estimation des matrices AR à l’aide des données d’entraînement. Dans cette publication, les auteurs proposent une estimation récursive fondée sur un filtrage de Kalman et ses variantes. Une étude comparative des différentes méthodes est menée sur les données fournies par la DGA – Maîtrise de l’Information.
multichannel autoregressive process, Parametric Adaptive Matched Filter (PAMF), Kalman filter, extended Kalman filter, sigma point Kalman filter, unscented Kalman filter, central difference Kalman filter.
MOTS-CLÉS
processus autorégressif vectoriel, Parametric Adaptive Matched Filter (PAMF), filtrage de Kalman, filtrage de Kalman étendu, filtrage de Kalman par sigma point, filtrage de Kalman non parfumé, filtrage de Kalman à différence centrale.
Abramovich Y. I., Johnson B. A., Spencer N. K., (2008). Two-Dimensional Multivariate Parametric Models for Radar Applications – Part I: Maximum-Entropy Extensions for Toeplitz-Block-Matrices. IEEE Transactions on Signal Processing, vol. 56, n. 11, pp. 5509-5526.
Alspach D. L. and Sorenson H. W., (1972). Nonlinear Bayesian Estimation Using Gaussian Sum Approximations. IEEE Transactions on Automatic Control, vol. 17, n. 4, pp. 439-448.
Arasaratnam I., Haykin S., Elliott R., (2007). Discrete-Time Nonlinear Filtering Algorithms Using Gauss-Hermite Quadrature. Proceedings of the IEEE, vol. 95, n. 5, pp. 953-977.
Belkacemi H. et Marcos S., (2006). Fast Iterative Subspace Algorithms for Airborne STAP Radar. EURASIP Journal on Advances in Signal Processing, Special Issue on Multisensor Processing for Signal Extraction and Applications, pp. 1-8.
Bidon S., Montécot M., Savy L., (2011). Introduction au STAP. Partie III : les données du club STAP. Revue Traitement du Signal, 2011.
Burg J. P., (1967). Maximum Entropy Spectral Analysis. Proceedings of the 37th Meeting Soc. of Exploration Geophysicists.
Cadwell J., Hale T., (2004). Parametric Adaptive Matched Filter Applied to Forward Looking Radar. Proceedings of the International Conference on Radar Systems, Toulouse, France.
Doucet A., De Freitas J., Gordon N., (2001). Sequential Monte Carlo in Practice, Cambridge, U. K., Cambriddge University Press.
Favier G., (2004). Signaux aléatoires : modélisation, estimation, detection, Chapitre 7 : Estimation paramétrique de modèles entrée-sortie. Ed. M. Guglielmi, Hermes, Lavoisier.
Guerci J. R. et Baranoski E. J., (2006). Knowledge-Aided Adaptive Radar at DARPA – an Overview. IEEE Signal Processing Magazine, vol. 23, n. 1.
Grewal M. S., Kain J., (2010). Kalman Filter Implementation With Improved Numerical Properties. IEEE Transactions on Automatic Control, vol. 55, no. 9, pp. 2058-2068.
Haimovitch A., (1996). Eigencanceler: Adaptive Radar by Eigenanalysis Methods. IEEE Transactions on Aerospace and Electronics Systems, vol. 32, n. 2, pp. 532-542, 1996.
Hasan K., Hossain J., Haque A., (2003). Parameter Estimation of Multichannel Autoregressive Processes in Noise. Signal Processing, vol. 83, pp. 603-610.
Haykin S., Kesler S., Currie B., (1979). An Experimental Classification of Radar Clutter. Proceedings of the IEEE, vol. 67, pp. 332-333.
Ito K., Xiong K., (2000). Gaussian Filters for Nonlinear Filtering Problems. IEEE Transactions on Automatic Control, vol. 5, pp. 910-927.
Julier S. J., Uhlmann J. K., Durrant-Whyte H. F., (1995). A New Approach for Filtering Nonlinear Systems. Proceedings of ACC, Seattle, WA, pp. 1628-1632.
Klemm R., (2004). Applications of Space-Time Adaptive Processing. IEE Radar, Sonar and Navigation and Avionics series 14, London, UK.
Lefebvre T., Bruyninckx H., De Schutter J., (2002). Comment on ‘A New Method for the Nonlinear Tranformation of Means and Covariances in Filters and Estimators’. IEEE Transactions on Automatic Control, vol. 47, n. 8, pp. 1406-1408.
Li W. and Jia Y., (2010). H∞ Filtering for a Class of Nonlinear Discrete-Time Systems based on Unscented Transform. Signal Processing, vol. 90, pp. 3301-3307.
Lombardo P., Colone F., (2003). Non-Linear STAP Filters Based on Adaptive 2D-FIR Filters. Proceedings of IEEE Radar Conference, pp. 51-58, Huntsville, AL.
Mahmoudi A., Karimi M., (2008). Estimation of the Parameters of Multichannel Autoregressive Signals from Noisy Observations. Signal Processing, vol. 88, n. 11, pp. 2777-2783.
Mahmoudi A., Karimi M., (2010). Parameter Estimation of Autoregressive Signals from Observations Corrupted with Colored Noise. Signal Processing, vol. 90, n. 1, pp. 157-164.
Marple S. L., (1987). Digital Analysis with Applications. Prentice Hall Signal Processing Series.
Michels J. H., (1991). Multichannel Detection Using the Discrete-time Model-Based Innovations Approach. Technical Report of Rome Laboratory.
Michels J. H., Rangaswamy M., Himed B., (2000). Performance of STAP Tests in Compound- Gaussian Clutter. Proceedings of the IEEE Sensor Array and Multichannel Signal Processing Workshop, pp. 250-255, Cambridge, MA.
Michels J. H., Himed B., Rangaswamy M., (2000). Evaluation of the Normalized Parametric Adaptive Matched Filter STAP Test in Airborne Radar Clutter. Proceedings of IEEE International Radar Conference, pp. 769-774, Alexandria, VA.
Michels J. H., Himed B., Rangaswamy M., (2000). Performance of Parametric and Covariance Based STAP Tests in Gaussian and Compound-Gaussian Clutter. Digital Signal Processing, vol. 10, n. 4, pp. 309-324.
Morf M., Dickinson B., Kailath T.,Viera A., (1977). Efficient Solution of Covariance Equation for Linear Prediction. IEEE Transactions on Acoustics Speech and Signal Processing, vol. ASSP-25, pp. 429-433.
Najim M., (2006). Modélisation, Estimation et Filtrage Optimal en Traitement du Signal. Hermes Science Publications.
Norgaard M., Poulsen N. K., Ravn O., (2000). New Developments in State Estimation for Nonlinear Systems. Automatica, vol. 36, pp. 1627-1638.
Nuttall A. H., (1976). Multivariate Linear Predictive Spectral Analysis Employing Weighted Forward and Backward Averaging: a Generalization of Burg’s Algorithm. Technical Report, Naval Underwater Systems Center.
Ovarlez J.-P., Pascal F., Forster P., Ginolhac G., Mahot M., (2011b). Traitement STAP et modélisation SIRV : Robustesse et persymétrie. Revue Traitement du Signal, 2011.
Parker P., Swindlehusrt A., (2003). Space-Time Autoregressive Filtering for Matched Subspace STAP. IEEE Transactions on Aerospace and Electronic Systems, vol. 39, n. 2, pp. 510-520.
Petitjean J., (2010). Contributions au traitement spatio-temporel fondé sur un modèle autorégressif vectoriel des interférences pour améliorer la détection de petites cibles lentes dans un environnement de fouillis hétérogène Gaussien et non Gaussien. Directeur de Thèse : Eric Grivel. Thèse de doctorat soutenue à l’Université de Bordeaux 1.
Petitjean J., Grivel E., Bobillet W., Roussilhe P., (2008). Multichannel AR Parameter Estimation form Noisy Observations as an Errors-In-Variables Issue. Proceedings of EURASIP-EUSIPCO 2008, Lausanne, Switzerland.
Petitjean J., Grivel E., Roussilhe P., (2009). H∞ Filtering for Autoregressive Modeling Based Space-Time Adaptive Processing. Proceedings of EURASIP-EUSIPCO 2009, Glasgow, Scotland.
Rangaswamy M., Mitchels J. H., (1997). A Parametric Multichannel Detection Algorithm for Correlated non-Gaussian Random Processes. Proceedings of IEEE National Radar Conference, Syracuse, NY.
Roman J., Rangaswamy M., Davis D. W., Qingwen Z., Himed B., J. Michels, (2000). Parametric Adaptive Matched Filter for Airborne Radar Applications. IEEE Transactions on Aerospace and Electronic Systems, vol. 36, n. 2, pp. 677-697.
Strand O. N., (1977). Multichannel Complex Maximum Entropy (Auto-Regressive) Spectral Analysis. IEEE Transactions on Automatic Control, vol. 22, pp. 634-640.
Therrien C. W., (1992). Discrete Random Signals and Statistical Signal Processing. Prentice Hall.
Wan E. A., Van Der Merwe R., (2001). The Unscented Kalman Filter. Chapter 7 in Kalman Filtering and Neural Networks. Adaptive and Learning Systems for Signal Processing, Communications, and Control, Wiley.
Ward J., (1994). Space-Time Adaptive Processing for Airborne Radar. Technical Report 1015, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA.
Wensink H. E., (2000). On Parametric Detection of Small Targets in Sea Clutter. Proceedings of the International Conference on Information Fusion, vol. 1, Paris, France.
Wiggins R., E. Robinson, (1965). Recursive Solution to the Multichannel Filtering Problem. Journal of Geophysical Research, vol. 70, n. 8, pp. 1885-1891.
Zheng W. X., (2000). Autoregressive Parameter Estimation from Noisy Data. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 47, n. 1, pp. 71-75.