PFE to Real, Second-Order SectionsWhen 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:
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.
Inverting the Z Transform