Converting the difference equation
to a *series* bank of real first- and second-order
sections is comparatively easy. In this case, we do not need a full
blown partial fraction expansion. Instead, we need only factor the
numerator and denominator of the transfer function into first- and/or
second-order terms. Since a second-order section can accommodate up
to two poles and two zeros, we must decide how to group pairs of poles
with pairs of zeros. Furthermore, since the series sections can be
implemented in any order, we must choose the section ordering. Both
of these choices are normally driven in practice by numerical
considerations. In fixed-point implementations, the poles and zeros
are grouped such that dynamic range requirements are minimized.
Similarly, the section order is chosen so that the intermediate
signals are well scaled. For example, internal overflow is more likely
if all of the large-gain sections appear before the low-gain sections.
On the other hand, the signal-to-quantization-noise ratio will
deteriorate if all of the low-gain sections are placed before the
higher-gain sections. For further reading on numerical considerations
for digital filter sections, see, *e.g.*, [103].

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