Line 16: | Line 16: | ||

left 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> | ||

+ | |||

+ | both sided: <math>x[n]</math> such that <math>x[n] = 0</math> ∀ n > n<sub>1</sub> and n < n<sub>2</sub> n<sub>1</sub> < n<sub>2</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> | 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> | ||

Line 51: | Line 53: | ||

::(i) includes the entire complex plane | ::(i) includes the entire complex plane | ||

::(ii) x[n] has no poles except maybe z = 0 or z = <math>\infty</math> | ::(ii) x[n] has no poles except maybe z = 0 or z = <math>\infty</math> | ||

− | ::(iii) | + | ::(iii) if for x[n], n<sub>1</sub> ≥ 0, z = <math>\infty</math> is included |

+ | ::(iv) if for x[n], n<sub>2</sub> ≤ 0, z = 0 is included | ||

+ | ::(v) if n<sub>1</sub> < 0 and n<sub>2</sub> > 0, then neither z = 0 or z = <math>\infty</math> is included |

## Revision as of 23:40, 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}

both sided: $ x[n] $ such that $ x[n] = 0 $ ∀ n > n_{1} and n < n_{2} n_{1} < n_{2}

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.

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| > z
_{0}(outside a circle in the z-plane) - (ii) includes $ \infty $ if n
_{0}≥ 0 - (iii) z = 0 could possibly be included in the ROC
- (iv) the farthest pole from the origin lies just inside |z
_{0}|

- (i) is generally defined by |z| > z
- (d) For Left Sided x[n], the ROC of X(z):
- (i) is generally defined by |z| < z
_{0}(inside a circle in the z-plane) - (ii) includes z = 0 if n
_{0}≤ 0 - (iii) x = $ \infty $ could possibly be included
- (iv) the nearest pole to the origin lies just outside of |z
_{0}|

- (i) is generally defined by |z| < z
- (e) For Both Sided x[n], the ROC of X(z):
- (i) is generally a ring in the z-plane such as z
_{1}< |z| < z_{2} - (ii) does not include z = 0 or z = $ \infty $

- (i) is generally a ring in the z-plane such as z
- (f) For finite duration x[n] (neither sided), the ROC of X(z):
- (i) includes the entire complex plane
- (ii) x[n] has no poles except maybe z = 0 or z = $ \infty $
- (iii) if for x[n], n
_{1}≥ 0, z = $ \infty $ is included - (iv) if for x[n], n
_{2}≤ 0, z = 0 is included - (v) if n
_{1}< 0 and n_{2}> 0, then neither z = 0 or z = $ \infty $ is included