Method

As can be seen from the code listing, this implementation of residuez simply calls residue, which was written to carry out the partial fraction expansions of -plane (continuous-time) transfer functions :

where is the quotient'' and is the remainder'' in the PFE:
 (J.1)

where is the order of the quotient polynomial in , and is the multiplicity of the th pole. (When all poles are distinct, we have for all .) For , we define . In the discrete-time case, we have the -plane transfer function
 (J.2)

For compatibility with Matlab's residuez, we need a PFE of the form such that

where . We see that the -plane case formally does what we desire if we treat -plane polynomials as polynomials in instead of . From Eq.(J.2), we see that this requires reversing the coefficient-order of B and A in the call to residue. In the returned result, we obtain terms such as

where the second form is simply the desired canonical form for -plane PFE terms. Thus, the th pole is

and the th residue is

Finally, the returned quotient polynomial must be flipped for the same reason that the input polynomials needed to be flipped (to convert from left-to-right descending powers of [] in the returned result to ascending powers of ).
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Example with Repeated Poles
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