### PFE to Real, Second-Order Sections

When all coefficients of and are real (implying that is the transfer function of 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:

*polar form*as , and the residue as , the last expression above can be rewritten as

*biquads*.

^{7.6}However, the full generality of a biquad section (two poles and two zeros) is not needed because the PFE requires only one zero per second-order term. 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*.

^{7.7}Thus, every strictly proper transfer function (with distinct poles) can be implemented using a parallel bank of two-pole, one-zero filter sections.

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