Szegö polynomials associated with Wiener-Levinson filters

摘要

Szegö polynomials are studied in connection with Wiener–Levinson filters formed from discrete signals xN={xN(k)}N−1k=0. Our main interest is in the frequency analysis problem of finding the unknown frequencies ωj, when the signal is a trigonometric polynomial xN(k)=∑j=−IIαjeiωjk. Associated with this signal is the sequence of monic Szegö polynomials {ρn(ψN; z)}∞n=0 orthogonal on the unit circle with respect to a distribution function ψN(θ). Explicit expressions for the weight function ψ′N(θ) and associated Szegö function DN(z) are given in terms of the Z-transform XN(z) of the signal xN. Several theorems are given to support the following conjecture which was suggested by numerical experiments: As N and n increase, the 2I + 1 zeros of ρn(ψN; z) of largest modulus approach the points eiωj. We conclude by showing that the reciprocal polynomials ≔ρ∗n(ψN; z)≔znρn(ψN;1z) are Padé numerators for Padé approximants (of fixed denominator degree) to a meromorphic function related to DN(z).