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

In the discrete-time case, we have the
-plane transfer function

For compatibility with Matlab's

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University