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

Abstract

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.1 β=0,Hence, in Engineering terminology Φ(t) (and often F (z)) is referred to as Finite Impulse Response (i.e. the support of Φ(t) is finite).Moreover F (z) will be called causal whenever 1 ≥ q (i.e. Φ(t) ≡ 0 for all 0 > t) and strictly causal if 0 ≥ q. Similarly, (strictly) anti-causal when (q ≥ n + 1) q ≥ n.

Characterizations of rectangular (para)-unitary rational functions

“…and by formally adding zero matrices to (5.1), we shall hereafter use the following MFD, where the numerator and denominator polynomials have the same power, By construction, both HÑ H * Ñ and H∆H * ∆ are of the same dimensions p(β + 1) × p(β + 1). 5 In principle, for arbitrary m × m-valued polynomial R(z), another RMFD is…”

“…This work is devoted to p × m-valued rational functions within U. In [5] we focused on the subset of (possibly Laurent) polynomials (within U) i.e.…”

“…In [5] we focus on the subclass rational functions: In mathematical terms F (z) are p×m-valued polynomials with powers of possibly mixed signs, i.e. where in (1.1) the matrix A is nilpotent (i.e.…”

“…We there present three characterizations of those functions within U. Here, (in Theorem 5.2 below) we use Theorem 5.1 to offer an alternative proof of one of the main results in [5].…”

Realizations of Infinite Products, Ruelle Operators and Wavelet Filters

“…An earlier relevant result on wavelet filter (1.1), (1.2) appeared in [11]. In [3] and [4], we recently further explored rectangular rational functions which are (co)isometric on the unit circle (with poles anywhere, but T).…”