Z Transform and LTI System Properties Study Guide

Introduction

This page will go over some of my conclusions about properties of the z-transform and discuss some examples of how they may be used to draw conclusions about LTI systems and digital filters. This topic assumes some basic knowledge of the z-transform and signal processing, however some definitions are provided below as a refresher or for reference. The information provided is intended to clarify or expand on some of the z-transform properties presented in class. Some of the information includes statements or arguments to validate or clarify claims; these are intended to aid understanding but not substitute for a proof.

definitions

z-transform: $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$ z ∈ ℂ

DTFT: $X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n}$

right sided: $x[n]$ such that $x[n] = 0$ ∀ n < n0

left sided: $x[n]$ such that $x[n] = 0$ ∀ n > n0

neither sided: $x[n]$ has finite duration, meaning $x[n] = 0$ ∀ n < n1 and n > n2, n1 < n2

Absolutely Summable: $x[n]$ is absolutely summable if: $\sum_{n=-\infty}^{\infty} |x[n]| = c$, where $c$ is a constant.

Relationship Between Fourier Transform and Z Transform

In the z-transform definition above, z is any complex number, which can be represented in polar coordinates by $z = re^{j\omega}$ Then the x-transform can be written as: $X(z) = X(re^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n](re)^{-j \omega n}$. Thus is r = 1, meaning that z is constrained to z = $e^{j\omega}$, then the z-transform is equivalent to the DTFT, and the DTFT is the z-transform constrained with z constrained to the unit circle in the z-plane.

Since the z-transform exists outside the unit circle, it is useful for analyzing signal that don't have a DTFT, analyzing LTI system stability, and looking at the transfer function characteristics among other uses.

Summary of Z Transform Properties

(a) The DTFT(x[n]) converges if the ROC of X(z) contains the unit circle on the z-plane.
(b) No poles of X(z) are included in the ROC of X(z), poles of X(z) shape the ROC.
(c) For Right Sided x[n], the ROC of X(z):
(i) is generally defined by |z| > z0 (outside a circle in the z-plane)
(ii) includes $\infty$ if n0 ≥ 0
(iii) z = 0 could possibly be included in the ROC
(iv) the farthest pole from the origin lies just inside |z0|
(d) For Left Sided x[n], the ROC of X(z):
(i) is generally defined by |z| < z0 (inside a circle in the z-plane)
(ii) includes z = 0 if n0 ≤ 0
(iii) x = $\infty$ could possibly be included
(iv)

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