The general second-order case with

(the so-called

*biquad* section) can be written when

as

To perform a

partial fraction expansion, we need to extract an order 0
(length 1) FIR part via

long division. Let

and rewrite

as a ratio of polynomials in

:

Then long division gives

yielding

or

The delayed form of the partial fraction expansion is obtained by
leaving the coefficients in their original order. This corresponds
to writing

as a ratio of polynomials in

:

Long division now looks like

giving

Numerical examples of partial fraction expansions are given in §

6.8.8
below. Another worked example, in which the

filter
is converted to a set of parallel, second-order
sections is given in §

3.12. See also §

9.2 regarding
conversion to second-order sections in general, and §

G.9.1 (especially
Eq.

(

G.22)) regarding
a

state-space approach to partial fraction expansion.

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