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Z-Transform

  • Frédéric Cohen Tenoudji
Chapter
Part of the Modern Acoustics and Signal Processing book series (MASP)

Abstract

The z-transform plays for digital signals the role of the Laplace transform for analog signals. The circumference of the zero centered circle of radius 1 in the z-plane plays the role of the imaginary axis of frequencies in the Laplace plane. In this chapter, after having defined the z-transform and the Fourier transform of a numerical sequence, we specify the domain of convergence of the power series, that is to say, the domain of definition of the z-transform. It is shown that the domain of convergence of a causal sequence is the exterior of a disc centered at the origin. We discuss the existence of the Fourier transform of a sequence.  Using the properties of integration on a closed contour and the residue theorem in the complex plane, we demonstrate the inversion formula of the z-transform. We show that the z-transform of a product of two series is given by a convolution formula in the frequency domain. Various properties of the z-transform are given in a table.

Keywords

Frequency Response Complex Plane Imaginary Axis Digital Filter Inversion Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Pierre and Marie Curie University, UPMCParisFrance

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