When all coefficients of and are real (implying that
is a realfilter), 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.^{7.4}However, 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}