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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