### Series, Real, Second-Order Sections

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].

#### Scaling:

The amplitude of the output is proportional to the amplitude of the input (thescaling property).

#### Superposition:

When two signals are added together and fed to the filter, the filter output is the same as if one had put each signal through the filter separately and then added the outputs (thesuperposition property).

While the implications of linearity are far-reaching, the mathematical
definition is simple. Let us represent the general *linear*
(but possibly *time-varying*) filter as a *signal operator*:

where is the entire input signal, is the output at time , and is the filter expressed as a

*real-valued function of a signal*for each . Think of the subscript on as selecting the th output sample of the filter. In general,

*each*output sample can be a function of several or even

*all*input samples, and this is why we write as the filter input.

**Definition. **A filter
is said to be
*linear*
if for any pair of signals
and for all
constant gains , we have the following relation for each
sample time
:

where denotes the signal space (complex-valued sequences, in general). These two conditions are simply a mathematical restatement of the previous descriptive definition.

The
*scaling* property of linear systems
states that scaling
the input of a linear system (multiplying it by a constant gain
factor) scales the output by the same factor.
The
*superposition* property of linear systems states that
the response of a linear system to a sum of signals is the sum of the
responses to each individual input signal. Another view is that the
individual signals which have been summed at the input are processed
independently inside the filter--they superimpose and do not
interact. (The addition of two signals, sample by sample, is like
converting stereo to mono by mixing the two channels together
equally.)

Another example of a linear signal medium is the earth's
atmosphere. When two sounds are in the air at once, the air pressure
fluctuations that convey them simply add (unless they are extremely
loud). Since any finite continuous signal can be represented as a sum
(*i.e.*, superposition) of sinusoids, we can predict the filter response to
any input signal just by knowing the response for all
sinusoids. Without superposition, we have no such general description
and it may be impossible to do any better than to catalog the filter
output for each possible input.

Linear operators distribute over linear combinations, *i.e.*,

#### Real Linear Filtering of Complex Signals

When a filter is a linear filter (but not necessarily time-invariant), and its input is a complex signal , then, by linearity,

Appendix H presents a linear-algebraic view of linear filters that can be useful in certain applications.

**Next Section:**

Why Dynamic Range Compression is Nonlinear

**Previous Section:**

Parallel Second-Order Signal Flow Graph