Efficient inverse $Z$-transform and Wiener-Hopf factorization

TL;DR

This paper introduces efficient numerical methods for inverting the Z-transform and performing Wiener-Hopf factorization using sinh-deformations, enabling fast and precise computations for probability moments and causal filters.

Contribution

It presents novel contour deformation techniques and simplified trapezoid rule approaches for rapid and accurate Z-transform inversion and Wiener-Hopf factorization.

Findings

Achieves precision of E-14 in microseconds

Achieves precision of E-11 in milliseconds

Applicable to probability moments and causal filter design

Abstract

We suggest new closely related methods for numerical inversion of $Z$-transform and Wiener-Hopf factorization of functions on the unit circle, based on sinh-deformations of the contours of integration, corresponding changes of variables and the simplified trapezoid rule. As applications, we consider evaluation of high moments of probability distributions and construction of causal filters. Programs in Matlab running on a Mac with moderate characteristics achieves the precision E-14 in several dozen of microseconds and E-11 in several milliseconds, respectively.