Statistical characterisation of the maximum eigenvalue of a wishart distribution with application to multi-channel SAR system

Multi-channel SAR system characterise the target with 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 characterise the target vector. In this framework, the eigendecomposition of the covariance matrix have demonstrated as a important analysis in 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. Related to this, the cornerstone study considering the statistical description of the covariance matrix eigendecomposition in polarimetry has been carried out in [1]. However, the majority of the analysis in [1] was performed on the basis of numerical methods. In this paper we support the results of [1] by addressing analytical solutions. Specifically, we derive new exact closed form expressions for Probability Density Function (PDF), for Cumulative Distribution Function (CDF) and for the Moment Generating Function (MGF) of the multi channel SAR system covari-ance matrix maximum eigenvalue, thus enabling the exact evaluation of the performance analysis of the estimation and the detection problem considering the number of averaged samples and different correlation scenario. Our results are analysed by means of simulated data.