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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 < 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.
