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==definitions== | ==definitions== | ||

− | z-transform: <math> X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} </math> | + | z-transform: <math> X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} </math> z ∈ ℂ |

+ | |||

+ | DTFT: <math> X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n} </math> | ||

right sided: <math>x[n]</math> such that <math>x[n] = 0</math> ∀ n < n<sub>0</sub> | right sided: <math>x[n]</math> such that <math>x[n] = 0</math> ∀ n < n<sub>0</sub> | ||

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Absolutely Summable: <math>x[n]</math> is absolutely summable if: <math> \sum_{n=-\infty}^{\infty} |x[n]| = c</math>, | Absolutely Summable: <math>x[n]</math> is absolutely summable if: <math> \sum_{n=-\infty}^{\infty} |x[n]| = c</math>, | ||

where <math>c</math> is a constant. | where <math>c</math> 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 | ||

+ | <math> z = re^{j\omega} </math> | ||

+ | Then the x-transform can be written as: | ||

+ | <math> X(z) = X(re^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n](re)^{-j \omega n} </math>. | ||

+ | Thus is r = 1, meaning that z is constrained to z = <math>e^{j\omega}</math>, 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. | ||

+ | |||

+ | ==Summary of Z Transform Properties== |

## Revision as of 20:59, 1 December 2019

## Contents

# 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 < n_{0}

left sided: $ x[n] $ such that $ x[n] = 0 $ ∀ n > n_{0}

neither sided: $ x[n] $ has finite duration, meaning $ x[n] = 0 $ ∀ n < n_{1} and n > n_{2}, n_{1} < n_{2}

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.