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==definitions==
 
==definitions==
  
z-transform: <math> X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} </math>
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z-transform: <math> X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}  </math> z &#8712; &#8450;
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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> &forall; n < n<sub>0</sub>
 
right sided: <math>x[n]</math> such that <math>x[n] = 0</math> &forall; 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.
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 +
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==Relationship Between Fourier Transform and Z Transform==
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In the z-transform definition above, z is any complex number, which can be represented in polar coordinates by
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<math> z = re^{j\omega} </math>
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Then the x-transform can be written as:
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<math> X(z) = X(re^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n](re)^{-j \omega n} </math>.
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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.
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==Summary of Z Transform Properties==

Revision as of 21:59, 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} $ 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.

Summary of Z Transform Properties

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett