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<math> X(z) = X(re^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n](re)^{-j \omega n} </math>. | <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. | 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. | ||
| + | |||
| + | 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== | ==Summary of Z Transform Properties== | ||
Revision as of 21:13, 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.
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.
