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Linear Filters

In everyday terms, the fact that a filter is linear means simply that the following two properties hold:

Scaling:

The amplitude of the output is proportional to the amplitude of the input (the scaling 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 (the superposition 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:

$\displaystyle y(n) = {\cal L}_n\{x(\cdot)\} \protect$ (5.2)

where $ x(\cdot)$ is the entire input signal, $ y(n)$ is the output at time $ n$, and $ {\cal L}_n\{\}$ is the filter expressed as a real-valued function of a signal for each $ n$. Think of the subscript $ n$ on $ {\cal L}_n\{\}$ as selecting the $ n$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 $ x(\cdot)$ as the filter input.


Definition. A filter $ {\cal L}_n$ is said to be linear if for any pair of signals $ x_1(\cdot),x_2(\cdot)$ and for all constant gains $ g$, we have the following relation for each sample time $ n\in{\bf Z}$:

$\displaystyle \hbox{Scaling:}\!\!$   $\displaystyle {\cal L}_n\{g\, x(\cdot) \} = g\,{\cal L}_n\{x(\cdot)\},
\quad\forall g\in{\bf C}, \;\forall x\in{\cal S}
\protect$ (5.3)
$\displaystyle \hbox{Superposition:}\!\!$   $\displaystyle {\cal L}_n\{x_1(\cdot) + x_2(\cdot)\}
= {\cal L}_n\{x_1(\cdot)\} + {\cal L}_n\{x_2(\cdot)\}$ (5.4)
    $\displaystyle \forall x_1,x_2\in{\cal S}
,
\protect$  

where $ {\cal S}$ 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.,

$\displaystyle \zbox {%
{\cal L}\{\alpha x_1 + \beta x_2\} = \alpha{\cal L}\{x_1\} + \beta {\cal L}\{x_2\}}
$

for any linear operator $ {\cal L}\{\}$, any real or complex signals $ x_1,
x_2\in{\cal S}$, and any real or complex constant gain factors $ \alpha,\beta$.


Real Linear Filtering of Complex Signals

When a filter $ {\cal L}_n\{x\}$ is a linear filter (but not necessarily time-invariant), and its input is a complex signal $ w \isdeftext x+jy$, then, by linearity,

$\displaystyle {\cal L}_n\{w\} \isdef {\cal L}_n\{x+jy\} = {\cal L}_n\{x\}+j{\cal L}_n\{y\}.
$

This means every linear filter maps complex signals to complex signals in a manner equivalent to applying the filter separately to the real and imaginary parts (which are each real). In other words, there is no ``interaction'' between the real and imaginary parts of a complex input signal when passed through a linear filter. If the filter is real, then filtering of complex signals can be carried out by simply performing real filtering on the real and imaginary parts separately (thereby avoiding complex arithmetic).

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


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Time-Invariant Filters
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Examples of Digital Filters