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Statistical Characterisation of the Maximum Eigenvalue of a Wishart Distribution with Application to Multi-Channel SAR System

Esra Erten(1,2), Rafael Zandona-Schneider(1), Andreas Reigber(1) and Olaf Hellwich(2)

(1) German Aerospace Center, DLR, D-82234 Wessling, Germany
(2) Technical University Berlin, Sekr. FR 3-1, Franklinstr. 28/29, 10587 Berlin, Germany


Multi-channel SAR systems characterize a target by a multicomponent Gaussian circular vector, whose number of components $m$ is equal to the number of polarimetric and/or interferometric channels of the system. In the case of the multivariate (multi-channel) Gaussian system, the second order statistics, known as covariance matrix, contains all the necessary information to characterize the target vector. In this framework, the eigendecomposition of the covariance matrix has been demonstrated as a important analysis for the physical parameter estimation and target detection. Especially, the maximum eigenvalue related to the first eigenvector of the covariance matrix is the most interesting parameter in a wide selection of application, i.e. polarimetry, GMTI (ground moving target indication) and interferometric phase filtering. In this paper we investigate the behavior of the maximum eigenvalue extracted from the eigendecomposition of the multi-channel covariance matrix of SAR system. In particular, we derive a closed-form expression of the probability density function, cumulative density function and the moment generating function of the maximum eigenvalue, thus enabling the exact evaluation of the estimation and the detection problem of the physical parameter considering the number of averaged samples and different correlation scenario. Our results are analyzed by means of simulated data to demonstrate in detail the effects of the bias in parameter estimation from multichannel system.


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