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Practical Signals Theory with MATLAB Applications by Richard J. Tervo

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The Illustrated z-Transform

In the pages that follow, a series of z-transform pairs are presented, each showing a discrete function h[n] on the left and the corresponding pole-zero diagram for H(z) on the right, where:

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Each of the h[n] may be considered as the impulse response of a discrete LTI system. In each pole-zero diagram, poles are shown with an X while zeros are plotted with an O. The region of convergence shown shaded in each pole-zero diagram is generally the entire z-plane outside of a circle (centered on the origin) that encompasses all the poles.

The inverse z-transform h[n] is only generally found from the pole-zero diagram of H(z) as constant values or gain factors are not visible when poles and zeros are plotted. Similarly, the sum of two pole-zero diagrams is not generally the sum of the corresponding z-transforms. Because the z-transform takes place on discrete (sampled) signals, the angular position of poles and zeros around the origin varies with sampling rate for a given frequency component.

The importance of the pole-zero diagram to system stability can also be seen, where poles outside the unit circle are associated with divergent impulse response functions. If the region of convergence includes the circle r = 1, then the related discrete time Fourier transform ...

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