# ch06 - Poularikas, A.D. The Z-Transform. The Transforms and...

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Poularikas, A.D . “ The Z- Transform .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000

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© 2000 by CRC Press LLC 6 The Z-Transform 6.1 Introduction A. One-Sided Z-Transform 6.2 The Z-Transform and Discrete Functions 6.3 Properties of the Z-Transform Linearity • Shifting Property • Time Scaling • Periodic Sequence • Multiplication by n and nT • Convolution • Initial Value • Final Value • Multiplication by ( nT ) k • Initial Value of f ( nT ) • Final Value for f ( nT ) • Complex Conjugate Signal • Transform of Product • Parseval’s Theorem • Correlation • Z- Transforms with Parameters 6.4 Inverse Z-Transform Power Series Method • Partial Fraction Expansion • Inverse Transform by Integration • Simple Poles • Multiple Poles • Simple Poles Not Factorable • F ( z ) is Irrational Function of z B. Two-Sided Z-Transform 6.5 The Z-Transform 6.6 Properties Linearity • Shifting • Scaling • Time Reversal • Multiplication by nT • Convolution • Correlation • Multiplication by e anT Frequency Translation • Product • Parseval’s Theorem • Complex Conjugate Signal 6.7 Inverse Z-Transform Power Series Expansion • Partial Fraction Expansion • Integral Inversion Formula C. Applications 6.8 Solutions of Difference Equations with Constant Coefﬁcients 6.9 Analysis of Linear Discrete Systems Transfer Function • Stability • Causality • Frequency Characteristics • Z-Transform and Discrete Fourier Transform (DFT) 6.10 Digital Filters Inﬁnite Impulse Response (IIR) Filters • Finite Impulse Responses (FIR) Filters 6.11 Linear, Time-Invariant, Discrete-Time, Dynamical Systems 6.12 Z-Transform and Random Processes Power Spectral Densities • Linear Discrete-Time Filters • Optimum Linear Filtering 6.13 Relationship Between the Laplace and Z-Transform Alexander D. Poularikas University of Alabama in Huntsville
© 2000 by CRC Press LLC 6.14 Relationship to the Fourier Transform Appendix: Tables 1 to 5 Table 1: Z-Transform Properties of the Positive-Time Sequences • Table 2: Z-Transform Properties for Positive- and Negative-Time Sequences • Table 3: Inverse Transform of the Partial Fractions of F ( z ) • Table 4: Inverse Transform of the Partial Fractions of F i ( z ) • Table 5: Z-Transform Pairs 6.1 Introduction The Z-transform is a powerful method for solving difference equations and, in general, to represent discrete systems. Although applications of Z-transforms are relatively new, the essential features of this mathematical technique date back to the early 1730s when DeMoivre introduced the concept of a generating function that is identical with that for the Z-transform. Recently, the development and extensive applications of the Z-transform are much enhanced as a result of the use of digital computers.

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## This note was uploaded on 12/03/2009 for the course EEE transforn taught by Professor Profcenk during the Spring '09 term at Dogus University.

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ch06 - Poularikas, A.D. The Z-Transform. The Transforms and...

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