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6.3 Linear Time-Invariant Systems 121

we could design a filter that has the capability to suppress the portion of the
signal with a periodicity of τ=15, whereas the other two cycles are unaf-
fected. Such simple periodic signals can also be used to predict signal distor-
tions of natural fi lters.
A step function is another basic input signal that can be used for exploring
filter characteristics. It describes the transition from a value of one towards

zero at a certain time.

t = (1:100)';
x = [ones(50,1);zeros(50,1)];
plot(t,x), axis([0 100 -2 2])

This signal can be used to study the effects of a filter on a sudden transi-

tion. An abrupt climate change could be regarded as an example. Most
natural filters tend to smooth such a transition and smear it over a longer

time period.
The unit impulse is the third important signal that we will use in the fol-
lowing examples. This signal equals zero for all times except for a single
data point which equals one.
t = (1:100)';
x = [zeros(49,1);1;zeros(50,1)];
plot(t,x),axis([0 100 -4 4])

The unit impulse is the most popular synthetic signal for studying the per-
formance of a filter. The output of the filter, the impulse response, describes

the characteristics of a filter very well. Moreover, the output of a linear time-

invariant filter can be described by the superposition of impulse responses

that haven been scaled by the amplitude of the input signal.

6.3 Linear Time-Invariant Systems

Filters can be described as systems with an input and output. We therefore
first describe the characteristics of a more general system before we proceed

to apply this theory to filters. Important characteristics of a system are
1. Continuity – A system with continuous inputs and outputs is continuous.
Most of the natural systems are continuous. However, after sampling na-
tural signals we obtain discrete data series and model these natural sy-
stems as discrete systems, which have discrete inputs and outputs.