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We here study finite impulse response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class $$\mathcal {U}$$U). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in $$\mathcal {U}$$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 $$\mathcal {U}$$U, so is the whole family. A key role is played by Hankel matrices.

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