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} $

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.