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Example: The General Biquad PFE

The general second-order case with $ M=N=2$ (the so-called biquad section) can be written when $ b_0\ne 0$ as

$\displaystyle H(z) \eqsp g\frac{1 + b_1 z^{-1}+ b_2 z^{-2}}{1 + a_1 z^{-1}+ a_2 z^{-2}}.
$

To perform a partial fraction expansion, we need to extract an order 0 (length 1) FIR part via long division. Let $ d=z^{-1}$ and rewrite $ H(z)$ as a ratio of polynomials in $ d$ :

$\displaystyle H(d^{-1}) \eqsp g\frac{b_2 d^2 + b_1 d + 1 }{a_2 d^2 + a_1 d + 1}
$

Then long division gives % For typesetting long division --- NEEDED WITHIN THE MAKEIMAGE ENV?
% (raw TeX, not LaTeX)\begin{displaymath}
\begin{array}[t]{@{}rcccc@{}}
& & & \frac{b_2}{a_2} & \\ [5pt] %
\noalign{\moveright 9.5\digitwidth%
\vbox{\hrule width 22\digitwidth}}
a_2~~a_1~~1~\bigg)\,
& b_2 & b_1 & 1 & \\
& b_2 & \frac{b_2}{a_2}a_1 & \frac{b_2}{a_2} &\\ [5pt] %
\noalign{\moveright 11\digitwidth%
\vbox{\hrule width 22\digitwidth}}\vspace{5pt}
& & b_1-\frac{b_2}{a_2}a_1 & 1-\frac{b_2}{a_2} &
\end{array}\end{displaymath}
yielding

$\displaystyle H(d^{-1}) \eqsp g\frac{b_2}{a_2} + g\frac{\left(b_1-\frac{b_2}{a_2}a_1\right)d+
\left(1-\frac{b_2}{a_2}\right)}{a_2d^2 + a_1d + 1}
$

or

$\displaystyle H(z) \eqsp g\frac{b_2}{a_2} +
g\frac{\left(1-\frac{b_2}{a_2}\right)
+\left(b_1-\frac{b_2}{a_2}a_1\right)z^{-1}}{1 + a_1z^{-1}+ a_2z^{-2}}.
$

The delayed form of the partial fraction expansion is obtained by leaving the coefficients in their original order. This corresponds to writing $ H(z)$ as a ratio of polynomials in $ z$ :

$\displaystyle H(z) \eqsp g\frac{z^2 + b_1 z + b_2 }{z^2 + a_1 z + a_2}
$

Long division now looks like % For typesetting long division --- NEEDED WITHIN THE MAKEIMAGE ENV?\begin{displaymath}
\begin{array}[t]{@{}rcccc@{}}
& & & 1 & \\ %
\noalign{\moveright 9.5\digitwidth%
\vbox{\hrule width 22\digitwidth}}
1~~a_1~~a_2~\bigg)\,
& ~1~~ &b_1 & b_2 & \\
& ~1~~ & a_1 & a_2 &\\ %
\noalign{\moveright 11\digitwidth%
\vbox{\hrule width 22\digitwidth}}
& & b_1-a_1 & b_2-a_2 &
\end{array}\end{displaymath}
giving

$\displaystyle H(z) \eqsp g + z^{-1}g\frac{(b_1-a_1) + (b_2-a_2)z^{-1}}{1 + a_1 z^{-1}+ a_2 z^{-2}}.
$

Numerical examples of partial fraction expansions are given in §6.8.8 below. Another worked example, in which the filter $ y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5)$ 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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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA