Characterizations of families of rectangular finite impulse response, para-unitary systems

TL;DR

This paper characterizes and parametrizes families of rectangular FIR systems that are para-unitary on the unit circle, providing new descriptions and construction methods for such systems using Hankel matrices.

Contribution

It offers three characterizations, a polytope-based parametrization, and six construction methods for FIR para-unitary systems, enhancing understanding and synthesis of these systems.

Findings

Three characterizations of FIR para-unitary systems

A polytope-based parametrization of all such FIRs

Six construction methods preserving para-unitarity

Abstract

We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class $\mathcal{U}$). First, we offer three characterizations of these systems. Then, introduce a %easy-to-use description of all FIRs in $\mathcal{U}$, as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs. Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in $\mathcal{U}$, so is the whole family. A key role is played by Hankel matrices.