Alternative Realizations

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

(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 . 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:

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 `' 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.

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Subsections

• First-Order Parallel Sections
• Parallel, Real, Second-Order Sections
• Parallel Second-Order Signal Flow Graph
• Series, Real, Second-Order Sections

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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.