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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 < 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.
