We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class U). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in 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 U, so is the whole family.
A key role is played by Hankel matrices.
D. Alpay, P. Jorgensen, and I. Lewkowicz. Characterizations of families of rectangular, finite impulse response, para-unitary systems. Journal of Applied Mathematics and Computing (2017) volume 54: 395-423.