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

ArXiv ID: 2404.19290 “View on arXiv”

Authors: Unknown

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.

Keywords: Z-transform Inversion, Wiener-Hopf Factorization, Numerical Methods, Probability Distributions, Causal Filters, Quantitative Methods (General)

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical techniques (sinh-acceleration, contour deformations, Wiener-Hopf factorization) with complex derivations and error bounds, but lacks backtesting, financial data analysis, or implementation-focused empirical results.

  flowchart TD
    A["Research Goal<br>Efficient numerical inversion<br>of Z-transform & Wiener-Hopf factorization"] --> B["Methodology<br>Sinh-contour deformation<br>& change of variables"]
    B --> C["Data/Input<br>Functions on the unit circle"]
    C --> D["Computational Process<br>Simplified trapezoid rule"]
    D --> E["Key Outcomes<br>1. Evaluation of high moments<br>2. Construction of causal filters<br>3. Precision: E-14 (μs) & E-11 (ms)"]