Graphics Programs Reference
In-Depth Information
we could design a fi lter that has the capability to suppress the portion of the
signal with a periodicity of
=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
fi lter characteristics. It describes the transition from a value of one towards
zero at a certain time.
τ
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 fi lter on a sudden transi-
tion. An abrupt climate change could be regarded as an example. Most
natural fi lters 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 fi lter. The output of the fi lter, the impulse response, describes
the characteristics of a fi lter very well. Moreover, the output of a linear time-
invariant fi lter 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
fi rst describe the characteristics of a more general system before we proceed
to apply this theory to fi lters. 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.
