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


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.


Frequency Response Complex Plane Imaginary Axis Digital Filter Inversion Formula 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Pierre and Marie Curie University, UPMCParisFrance

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