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 , , , and . 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.

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.