Asymptotic generalized eigenvalue distribution of Toeplitz block Toeplitz matrices

In many detection and estimation problems associated with processing of second order stationary 2-D discrete random processes, the observation data are the sum of two zero-mean second order stationary processes: the process of interest and the noise process. In particular, the main performance criterion is the signal to noise ratio (SNR). After linear filtering, the optimal SNR corresponds to the maximal value of a Rayleigh quotient which can be interpreted as the largest generalized eigenvalue of the covariance matrices associated with the signal and noise processes, which are Toeplitz block Toeplitz structured. In this paper, an extension of Szego´s theorem to the generalized eigenvalues of Hermitian Toeplitz block Toeplitz matrices is given, under the hypothesis of absolutely summable elements, providing information about the asymptotic distribution of those generalized eigenvalues and in particular of the optimal SNR after linear filtering.