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==Introduction== | ==Introduction== | ||

This page will go over some of my conclusions about properties of the z-transform and discuss some examples of how they may | 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. 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. | + | 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== | ==definitions== | ||

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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> | ||

+ | |||

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

+ | |||

+ | neither sided: <math>x[n]</math> has finite duration, meaning <math>x[n] = 0</math> ∀ n < n<sub>1</sub> and n > n<sub>2</sub>, n<sub>1</sub> < n<sub>2</sub> | ||

+ | |||

+ | 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. |

## Revision as of 21:15, 1 December 2019

# 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 < 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.