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124 6 Signal Processing

be delayed after filtering to compensate the differences between causal

and non-causal indexing.

6. Stability – A system is stable if the output of a finite input is also fi nite.
Stability is critical in filter design, where filters often have the disadvan-

tage of provoking diverging outputs. In such cases, the filter design has to
be revised and improved.
Linear time-invariant (LTI) systems as a special type are very popular. Such
systems have all the advantage that have been described above. They are
easy to design and to use in many applications. The following chapters 6.4
to 6.9 describe the design, realization and application of LTI-type fi lters to
extract certain frequency components of signals. These filters are mainly

used to reduce the noise level in signals. Unfortunately many natural sys-
tems do not behave as LTI systems. In many cases the signal-to-noise ratio
varies with time. Chapter 6.10 describes the application of adaptive fi lters
that automatically adjust their characteristics in a time-variable environ-
ment.

6.4 Convolution and Filtering

The mathematical description of a system transformation is convolution.
Filtering is one application of the convolution process. Running mean of
length five provides an example of such a simple filter. The output of an

arbitrary input signal is

The output y(t) is simply the average of the five input values x(t-2), x(t-1),
x(t), x(t+1) and x(t+2). In other words, all the fi ve consecutive input values
are multiplied by a factor of 1/5 and summed to form y(t). In this exam-
ple, all input values are multiplied by the same factor, i.e., they are equally
weighted. The fi ve factors used in the above operation are also called fi lter
weights b . The fi lter can be represented by the vector
k
b = [0.2 0.2 0.2 0.2 0.2]

consisting of the identical filter weights. Since this filter is symmetric, it

does not shift the signal on the time axis. The only function of this filter is to