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Understand the phenomenon of repeated poles in a Z-transform with parallel and series configurations. Learn how to handle repeated poles analytically and evaluate residues for a pole of multiplicity. Dive into an example of the impulse response of repeated poles.
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Lecture 5 The Z-Transform
Repeated Poles When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have In the series case, we get Thus, two one-pole filters in parallel are equivalent to a new one-pole filter7.8 (when the poles are identical), while the same two filters in series give a two-pole filter with a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms for a pole having multiplicity 2.
Dealing with Repeated Poles Analytically A pole of multiplicity mi has mi residues associated with it. For example, Let rik denote the jth residue associated with the pole pi , j=1,…,mi. Successively differentiating k-1 times with respect to z-1 and setting z=pi isolates the residue rik :
Example = r12 r13 r11 =
