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Concept# Z-transform

Summary

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-time equivalent of the Laplace transform (s-domain). This similarity is explored in the theory of time-scale calculus.
Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the discrete-time Fourier transform is evaluated over the unit circle of the z-domain. What is roughly the s-domain's left half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain.
One of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical app

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In this paper, we present a stable method for simulating passive distributed networks connected to linear and nonlinear devices using the time marching (TM) method. The method is based on the use of the Z-transform for analyzing and obtaining the TM equations, and the application of cepstrum for preventing the apparition of unstable poles in the Z-domain counterparts of the TM equations. The efficiency of the proposed method is demonstrated by comparing its results with simulations obtained using the following: 1) a full-wave FDTD approach, and 2) CST Cable Studio.

Shift-invariant spaces play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. A special class of the shift-invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points. One of the vital tools used in the study of sampling spaces is the Zak transform. The Zak transform is also related to the Poisson summation formula and a common thread between all these notions is the Fourier transform.

2012A stable method for the analysis of lossy transmission lines with nonlinear loads is presented. The simulation employs the time marching (TM) method for solving the convolution equations in the time domain. To cope with the late-time instability of the TM method, a novel approach based on the application of Z-transform analysis and cepstrum is presented. The efficiency of the proposed method is demonstrated by comparing its results with simulations obtained using (i) a full-wave FDTD approach, and (ii) CST Cable Studio.