# A Fourier series method for evaluation inverse Z-transforms

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## Abstract

Frequency-domain analysis of the sampling process shows that the Laplace transform of the sampler output is a periodic function of the complex variable s with period equal to j[omega][subscript s], where [omega][subscript s] = 2Ï€/T is the sampling frequency in radians per second and T is the sampling period in seconds. In addition to showing the periodic property of the sampler output, the analysis yields an infinite series representation for the Laplace transform of the sampler output in the variable e^sT. The coefficient of the n^th term of the series is the value of the time function at the n^th sampling instant. This infinite series representation is used to define the direct Z-transform by introducing a change of variables Z=e^sT. The inverse Z-transform is then obtained by evaluating the coefficients of the infinite series in Z. A method for evaluating these coefficients using Fourier series is presented in this thesis. Having established that the transform of the sampler output is periodic, it is shown that the transform of the output can be represented by a Fourier series in that region of the Z-plane where the convergence of the series is uniform. The region of uniform convergence is established as the Z-plane region outside of a circle centered at the origin that encloses all of the singularities of the Z-transform. The Fourier series is then obtained by transforming the infinite series representation of the transformed output into a trigonometric series in this region. When the real and imaginary terms of the Fourier series are collected it is seen that the general coefficients of the infinite series in Z and therefore the value of the time function at the sampling instants is contained in the general Fourier coefficient for the series of real terms. The time-domain response of the sampler output is then obtained by evaluating the Fourier coefficients for the series expansion of the real part of the Z-transformed output.