### 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:

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

Expanding a transfer function into a sum of second-order terms with
real coefficients gives us the filter coefficients for a parallel bank
of real second-order filter sections. (Of course, each real pole can
be implemented in its own real one-pole section in parallel with the
other sections.) In view of the foregoing, we may conclude that every
real filter with can be implemented as a parallel bank
of *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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