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:
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
For compatibility with Matlab's residuez, we need a PFE of the form such that
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
Example with Repeated Poles
The Padé-Prony Method