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Parallel Second-Order Signal Flow Graph

Figure 3.25 shows the signal flow graph for the implementation of our example filter using parallel second-order sections (with one first-order section since the number of poles is odd). This is the same filter as that shown in Fig.3.1 with $ g_1 = 0.5^3$, $ g_2 = 0.9^5$, $ M_1 = 3$, and $ M_2=5$. The second-order sections are special cases of the ``biquad'' filter section, which is often implemented in software (and chip) libraries. Any digital filter can be implemented as a sum of parallel biquads by finding its transfer function and computing the partial fraction expansion.

Figure 3.25: Signal flow graph for the re-implementation of the example filter $ y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5)$ as a parallel bank of real first- and second-order digital filter sections.
\begin{figure}\input fig/epfe.pstex_t

Figure: Same as Fig.3.25 except using transposed direct-form-II biquad sections in place of direct-form-II biquad sections.
\begin{figure}\input fig/epfet.pstex_t

The two second-order biquad sections in Fig.3.25 are in so-called ``Direct-Form II'' (DF-II) form. In Chapter 9, a total of four direct-form filter implementations will be discussed, along with some other commonly used implementation structures. In particular, it is explained there why Transposed Direct-Form II (TDF-II) is usually a better choice of implementation structure for IIR filters when numerical dynamic range is limited (as it is in fixed-point ``DSP chips''). Figure 3.26 shows how our example looks using TDF-II biquads in place of the DF-II biquads of Fig.3.25.

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