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Alternative Realizations

For actually implementing the example digital filter, we have only seen the difference equation

$\displaystyle y(n) = x(n) + g_1\, x(n-M_1) - g_2\, y(n-M_2)
$

(from Eq.$ \,$ (3.1), diagrammed in Fig.3.1). While this structure, formally known as ``direct form I'', happens to be a good choice for digital comb filters, there are many other structures to consider in other situations. For example, it is often desirable, for numerical reasons, to implement low-pass, high-pass, and band-pass filters as series second-order sections. On the other hand, digital filters for simulating the vocal tract (for synthesized voice applications) are typically implemented as parallel second-order sections. (When the order is odd, there is one first-order section as well.) The coefficients of the first- and second-order filter sections may be calculated from the poles and zeros of the filter.

We will now illustrate the computation of a parallel second-order realization of our example filter $ y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5)$ . As discussed above in §3.11, this filter has five poles and three zeros. We can use the partial fraction expansion (PFE), described in §6.8, to expand the transfer function into a sum of five first-order terms:

\begin{eqnarray*}
H(z) &=& \frac{1 + 0.5^3 z^{-3}}{1 + 0.9^5 z^{-5}}
\mathrel{=} \sum_{i=1}^5 \frac{r_i}{1-p_iz^{-1}}\\
&\approx&
\frac{0.1657}{1 + 0.9z^{-1}}
+
\frac{
0.1894 - j 0.0326
}{1 - (
0.7281 + j 0.5290
)z^{-1}}
+
\frac{
0.1894 + j 0.0326
}{1 - (
0.7281 - j 0.5290
)z^{-1}}
\\
&&
\qquad\qquad
\mathrel{+}
\frac{
0.2277 + j 0.0202
}{1 - (
-0.2781 + j 0.8560
)z^{-1}}
+ \frac{
0.2277 - j 0.0202
}{1 - (
-0.2781 - j 0.8560
)z^{-1}}\\ [5pt]
&=&
\frac{0.1657}{1+0.9000z^{-1}}
+
\frac{0.3788 -0.2413z^{-1}}{1 - 1.4562z^{-1}+ 0.8100z^{-2}}\\
&&
\qquad\qquad\qquad\;
\mathrel{+}
\frac{0.4555 + 0.0922z^{-1}}{1 + 0.5562z^{-1}+ 0.8100z^{-2}},
\end{eqnarray*}

where, in the last step, complex-conjugate one-pole sections are combined into real second-order sections. Also, numerical values are given to four decimal places (so `$ =$ ' is replaced by `$ \approx$ ' in the second line). In the following subsections, we will plot the impulse responses and frequency responses of the first- and second-order filter sections above.



Subsections
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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