12: Z-Transform and Discrete Time System Design
- Page ID
- 22920
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 12.1: Z-Transform
- The Z-transform is an extension of the Discrete-Time Fourier Transform (DTFT) used because DTFT often doesn't exist for many signals, whereas the Z-transform does. It simplifies notation as it uses a complex variable \(z\) instead of purely imaginary parameters. Although the Z-transform equations resemble DTFT equations, the Z-transform incorporates complex variable theory, beneficial in digital signal processing.
- 12.2: Common Z-Transforms
- The page provides a table summarizing common signals in continuous time Fourier series along with their corresponding Z-transforms and regions of convergence. Signals like the delta function, unit step, exponential, and trigonometric functions are listed, with their transforms expressed in terms of \(s\). The region of convergence varies depending on the expression, ranging from all real \(s\) to specific conditions based on parameters like \(\lambda\), \(a\), and \(b\).
- 12.3: Properties of the Z-Transform
- This module covers the basic properties of the Z-Transform for discrete-time signals and provides similarities to continuous-time and periodic signals. Key properties include linearity, symmetry, time scaling and shifting, convolution, and modulation. Each property helps simplify complex signal transformations. The module also highlights Parseval's Relation and the importance of frequency domain transformations for easier equation handling.
- 12.4: Inverse Z-Transform
- The page provides an overview of methods used to find the inverse of the z-transform, a key concept in digital signal processing. Four main techniques are discussed: Inspection, Partial-Fraction Expansion, Power Series Expansion, and Contour Integration. Each method is detailed with examples, highlighting their utility and application. A conclusion emphasizes the importance of the inverse z-transform in filter design.
- 12.5: Poles and Zeros in the Z-Plane
- This module will look at the relationships between the z-transform and the complex plane. Specifically, the creation of pole/zero plots and some of their useful properties are discussed.
- 12.6: Region of Convergence for the Z-Transform
- The document introduces the concept of the region of convergence (ROC) in the context of the z-transform, which is crucial for understanding when the output of a linear time-invariant (LTI) system converges. Key points include the ROC properties, which vary depending on whether the signal is finite-duration, right-sided, left-sided, or two-sided. The document also explains how to find the ROC and gives examples illustrating the calculation of the z-transform and determination of the ROC.
- 12.7: Rational Functions and the Z-Transform
- This module will introduce rational functions and describe some of their properties. In particular, it will discuss how rational functions relate to the z-transform and provide a useful tool for characterizing LTI systems.
- 12.9: Discrete Time Filter Design
- Describes how to design a general filter from the Z-Transform and it pole/zero plots.