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
- This page discusses the Z-transform, which generalizes the Discrete-Time Fourier Transform (DTFT) for signals where DTFT may not converge. It incorporates complex variables and includes Bilateral and Unilateral forms. The Z-transform is vital for analyzing digital filters and deriving transfer functions, providing an efficient approach to signal processing. Additionally, it offers a method for identifying systems and signals, with a demonstration available for interactive learning.
- 12.2: Common Z-Transforms
- This page presents a table illustrating common signals and their Z-transforms, detailing signals like the Dirac delta function, unit step function, and exponential functions. It includes Z-transform expressions and highlights the regions of convergence, which depend on various parameters such as \(s\) and \(\lambda\).
- 12.3: Properties of the Z-Transform
- This page covers essential properties of the Z-Transform for discrete-time signals, including linearity, symmetry, time scaling, time shifting, convolution, and time differentiation. It highlights the connection between time-domain and frequency-domain representations, showing how linear combinations and time shifts function in frequency.
- 12.4: Inverse Z-Transform
- This page outlines four methods for finding the inverse z-transform of \(X(z)\): Inspection, Partial-Fraction Expansion, Power Series Expansion, and Contour Integration. Each method is explained with examples, emphasizing the use of transform pair tables, rational function breakdown, power series coefficients, and complex variable theory. The importance of the inverse z-transform in engineering filter design is also highlighted.
- 12.5: Poles and Zeros in the Z-Plane
- This page explores poles and zeros in the context of the Z-transform, highlighting their importance in system analysis through graphical means. It defines zeros (where gain is zero) and poles (where gain is infinite) and utilizes the Z-plane for visualization. Examples demonstrate finding and plotting poles and zeros, including pole-zero cancellation.
- 12.6: Region of Convergence for the Z-Transform
- This page explains the region of convergence (ROC) related to the z-transform of discrete-time LTI systems, emphasizing that ROC indicates where the z-transform converges and cannot include poles. It varies by signal type (finite, right-sided, left-sided, or two-sided) and is illustrated through examples of two discrete-time signals.
- 12.7: Rational Functions and the Z-Transform
- This page introduces rational functions as quotients of polynomials and discusses their relevance to z-transforms. Key concepts such as roots, discontinuities, domain, and intercepts are defined, with an emphasis on the importance of roots in determining poles and zeros. The text concludes by highlighting that understanding rational functions is crucial for analyzing LTI systems and designing digital filters.
- 12.8: Difference Equations
- This page explores the significance of linear constant-coefficient difference equations (LCCDE) in digital signal processing (DSP), particularly for modeling linear time-invariant (LTI) systems. It covers the definitions, forms, and resolution methods for LCCDEs, including direct and indirect methods, especially dealing with multiple roots.
- 12.9: Discrete Time Filter Design
- This page explains the relationship between the z-transform, pole/zero plots, and frequency response in discrete-time systems. It details how frequency response magnitude is influenced by pole and zero positions in the z-plane, demonstrating amplitude variation from \(0\) to \(2\pi\). It emphasizes the importance of distances from the unit circle and behaviors surrounding poles and zeros, concluding that closeness to zeros reduces magnitude while closeness to poles increases it.