MIMO FIR equalizers and orders

Abstract

A necessary and sufficient condition is given for the existence of a polynomial left inverse for a polynomial (FIR) system. Additionally, random systems are defined, and a sufficient condition is given for almost sure existence of a polynomial inverse. The two results extend previous work in single-input, multiple-output systems to the case of multiple input systems and to functions of several variables. An algorithm to compute minimal order equalizers is also presented. Corollaries describe calibration of polarimetric, wide-band radar imagery.

Filter design for mimo sampling and reconstruction

“…Proof: The coprimeness condition on the minors guarantees the existence of an FIR such that and the periodicity conditions (15) and (16) hold. This is a standard result that can be proved using Bezout's identity [40]- [42]. Furthermore, letting , we also obviously have .…”

“…Thus, Theorem 2 generalizes all these problems simultaneously. In particular, when and all the discrete-time inputs are full-band signals, our result reduces to that of [42].…”

“…We point out some other relations of our results to existing results. The problem in [42] deals with existence of MIMO FIR equalizer filters in the absence of decimation of channel outputs, whereas the classical filter bank problem deals with a singleinput multiple-output channel whose outputs are decimated. The present problem deals with the existence of an MIMO FIR reconstruction filter in the presence of decimation.…”

“…The present problem deals with the existence of an MIMO FIR reconstruction filter in the presence of decimation. The solution depends not only on the channel transfer function matrix (as in [42] ) but on the decimation factor a well (as in the filter bank problem) and the band-structure of the inputs through .…”

“…Hence, the range space of equals the signal space for input , namely, . Now, it follows immediately that we can drop the constraint in (41) to obtain (42) where is the spectral norm of matrix for each , and is the largest singular value.…”