Two Methods for Numerical Inversion of the Z-Transform

TL;DR

This paper introduces two numerical methods, based on DFT and linear algebra, for inverting Z-transforms when analytical solutions are difficult, demonstrating their effectiveness through numerical examples.

Contribution

It proposes two simple, effective numerical methods for inverting Z-transforms, applicable to absolutely summable signals, with analysis of functions involving non-integer powers of z.

Findings

The methods are efficient and accurate as shown by numerical examples.

Functions with non-integer powers of z cannot be obtained from discrete-time signals.

The methods work under the condition of absolute summability, extendable via scaling.

Abstract

In some of the problems, complicated functions of the Z-transform variable, $z$, appear which either cannot be inverted analytically or the required calculations are quite tedious. In such cases numerical methods should be used to find the inverse Z-transform. The aim of this paper is to propose two simple and effective methods for this purpose. The only restriction on the signal (whose Z-transform is given) is that it must be absolutely summable (of course, this limitation can be removed by a suitable scaling). The first proposed method is based on the Discrete Fourier Transform (DFT) and the second one is based on solving a linear system of algebraic equations, which is obtained after truncating the signal whose Z-transform is known. Numerical examples are also presented to confirm the efficiency of the proposed methods. Functions in non-integer powers of $z$ are also briefly discussed and it is shown that such functions cannot be obtained by taking the Z-transform from any discrete-time signal.