Asymptotic Generalized Eigenvalue Distribution of Block Multilevel Toeplitz Matrices

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In many detection and estimation problems associated with processing of second-order stationary 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 block multilevel Toeplitz structured for <formula formulatype="inline"><tex Notation="TeX">$m$</tex></formula>-dimensional vector-valued second-order stationary <formula formulatype="inline"><tex Notation="TeX">$p$</tex> </formula>-dimensional random processes <formula formulatype="inline"><tex Notation="TeX">${\bf x}_{i_{1},i_{2},\ldots,i_{p}}\in \BBR ^{m}$</tex></formula>. In this paper, an extension of SzegÖ's theorem to the generalized eigenvalues of Hermitian block multilevel Toeplitz matrices is given, providing information about the asymptotic distribution of those generalized eigenvalues and in particular of the optimal SNR after linear filtering. A simple proof of this theorem, under the hypothesis of absolutely summable elements is given. The proof is based on the notion of multilevel asymptotic equivalence between block multilevel matrix sequences derived from the celebrated Gray approach. Finally, a short example in wideband space–time beamforming is given to illustrate this theorem. </para>