The z-Transform Notebook

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The forward rule base works for 1-D and m-D signals, whereas the inverse rule base only inverts 1-D and separable m-D transforms (<#1185#>Mathematica<#1185#> simply cannot perform the multiple contour integrations required in the general m-D case). Since the rule bases implement the bilateral form of the z-transform, <#699#>ZTransform<#699#> tracks the region of convergence (ROC) so that <#700#>InvZTransform<#700#> can uniquely determine the inverse.

The z-transform Notebook comes in three parts because of its size. Part I defines the bilateral z-transform and discusses the importance of the ROC in uniquely defining the transform. Discussion of the ROC leads naturally into introducing stability and causality of a signal based on the location of its poles relative to the ROC. Part II links the z-transform with the discrete-time Fourier transform (DTFT) and includes several animations relating pole-zero diagrams to magnitude frequency responses, a basic idea behind digital filter design [Oppenheim Schafer, 1989, ch.~5]. A figure in Part II, for example, shows one animated frame for a three-pole, two-zero filter whose magnitude response is being evaluated at a frequency of π rad. Scattered throughout the Notebook are several examples and exercises with solutions, many of which use examples of high-level digital signal analysis using <#701#>DSPAnalyze<#701#>, the discrete-time version of ASPAnalyze. Part III concludes the z-transform Notebook with two animations showing the deterioration of the magnitude response of a fourth-order elliptic filter when the location of the poles are slightly perturbed.