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:

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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