Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Svetlana Boyarchenko, Sergei Levendorskiĭ

TL;DR

The work addresses the challenge of numerically inverting the $Z$-transform and performing Wiener-Hopf factorization on the unit circle ${\mathbb T}$, where highly oscillatory and slowly decaying integrands hinder straightforward quadrature. It introduces conformal accelerations, notably sinh-acceleration and its variants (SINH1–SINH3) and LOG-acceleration, to deform contours and enable efficient, high-precision evaluation via the simplified trapezoid rule. The authors develop analytic bounds, parameter-choice guidelines, and practical algorithms for inverse $Z$-transform and WHF, including impulse-response construction for causal filters, with detailed examples on KoBoL distributions. Numerical experiments demonstrate substantial speedups and accuracy improvements, achieving error levels around $10^{-15}$ in microseconds to milliseconds, thus offering a robust toolkit for moment calculations and fast filtering in stochastic and signal-processing contexts.

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.

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

TL;DR

The work addresses the challenge of numerically inverting the -transform and performing Wiener-Hopf factorization on the unit circle , where highly oscillatory and slowly decaying integrands hinder straightforward quadrature. It introduces conformal accelerations, notably sinh-acceleration and its variants (SINH1–SINH3) and LOG-acceleration, to deform contours and enable efficient, high-precision evaluation via the simplified trapezoid rule. The authors develop analytic bounds, parameter-choice guidelines, and practical algorithms for inverse -transform and WHF, including impulse-response construction for causal filters, with detailed examples on KoBoL distributions. Numerical experiments demonstrate substantial speedups and accuracy improvements, achieving error levels around in microseconds to milliseconds, thus offering a robust toolkit for moment calculations and fast filtering in stochastic and signal-processing contexts.

Abstract

We suggest new closely related methods for numerical inversion of -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.
Paper Structure (15 sections, 8 theorems, 54 equations, 1 figure)

This paper contains 15 sections, 8 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Inverse $Z$-tranform and trapezoid rule
  3. Sinh-acceleration I
  4. General formulas and bounds
  5. Parameter choice
  6. Examples
  7. Sinh-acceleration II and III and Log-acceleration
  8. Sinh-acceleration II
  9. Sinh-acceleration III
  10. Log-acceleration
  11. Inverse $Z$-transform, Wiener-Hopf factorization of functions on ${\mathbb T}$, and construction of causal filters
  12. Wiener-Hopf factorization
  13. Calculation of the impulse response function
  14. Algorithm
  15. Conclusion

Key Result

Theorem 2.1

Let $h$ be analytic in ${\mathcal{D}}_{(1/\rho,\rho)}$, where $\rho>1$, and let the Hardy norm of $h$ be finite. Then

Figures (1)

  • Figure 1: (A) Solid line: 1101 nodes used in the trapezoid rule, dots: 33 nodes used in the Z-SINH-I algorithm. (B) Solid line: 901 nodes used in the trapezoid rule. Almost vertical line and dots: 306 and 55 nodes used in the Z-SINH-I and Z-SINH-II algorithm in Examples \ref{['ex:one-sidedKBL-moment']} and \ref{['ex:one-sidedKBL-moment-II']}, respectively.

Theorems & Definitions (25)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Lemma 3.2: stenger-book, Thm.3.2.1
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 4.1
  • ...and 15 more