PFE to Real, Second-Order Sections Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

PFE to Real, Second-Order Sections

When all coefficients of and are real (implying that is a real filter), it will always happen that the complex one-pole filters will occur in complex conjugate pairs. Let denote any one-pole section in the PFE of Eq. (6.7). Then if is complex and describes a real filter, we will also find somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:

Expressing the pole in polar form as , and the residue as , the last expression above can be rewritten as (see §10.1.3 for details)

In other terms, every real filter with can be implemented as a parallel bank of biquads.7.4However, the full generality of a biquad section is not needed, since the PFE requires only one zero per second-order section.

To see why we must stipulate in Eq. (6.7), consider the sum of two first-order terms by direct calculation:

 (7.9)

Notice that the numerator order, viewed as a polynomial in , is one less than the denominator order. In the same way, it is easily shown by mathematical induction that the sum of one-pole terms can produce a numerator order of at most (while the denominator order is if there are no pole-zero cancellations). Following terminology used for analog filters, we call the case a strictly proper transfer function.7.5

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