243
CHAPTER
13
Continuous Signal Processing
Continuous signal processing is a parallel field to DSP, and most of the techniques are nearly
identical.
For example, both DSP and continuous signal processing are based on linearity,
decomposition, convolution and Fourier analysis.
Since continuous signals cannot be directly
represented in digital computers, don't expect to find computer programs in this chapter.
Continuous signal processing is based on
mathematics
; signals are represented as equations, and
systems change one equation into another.
Just as the
digital computer
is the primary tool used
in DSP,
calculus
is the primary tool used in continuous signal processing.
These techniques have
been used for centuries, long before computers were developed.
The Delta Function
Continuous signals can be decomposed into scaled and shifted
delta functions
,
just as done with discrete signals.
The difference is that the continuous delta
function is much more complicated and mathematically abstract than its discrete
counterpart.
Instead of defining the continuous delta function by what it
is
, we
will define it by the
characteristics it has
.
A thought experiment will show how this works.
Imagine an electronic circuit
composed of linear components, such as resistors, capacitors and inductors.
Connected to the input is a signal generator that produces various shapes of
short
pulses
The output of the circuit is connected to an oscilloscope,
displaying the waveform produced by the circuit in response to each input
pulse.
The question we want to answer is:
how is the shape of the output
pulse related to the characteristics of the input pulse
?
To simplify the
investigation, we will only use input pulses that are much
shorter than the
output.
For instance, if the system responds in milliseconds, we might use input
pulses only a few microseconds in length.
After taking many measurement, we come to three conclusions:
First, the
shape
of the input pulse does not affect the shape of the output signal.
This

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244
is illustrated in Fig. 13-1, where various shapes of short input pulses
produce exactly the same shape of output pulse.
Second, the shape of the
output waveform is totally determined by the characteristics of the system,
i.e., the value and configuration of the resistors, capacitors and inductors.
Third, the
amplitude
of the output pulse is directly proportional to the
area
of the input pulse.
For example, the output will have the same amplitude
for inputs of:
1 volt for 1 microsecond, 10 volts for 0.1 microseconds,
1,000 volts for 1 nanosecond, etc.
This relationship also allows for input
pulses with
negative
areas.
For instance, imagine the combination of a 2
volt pulse lasting 2 microseconds being quickly followed by a -1 volt pulse
lasting 4 microseconds.
The total area of the input signal is
zero
, resulting
in the output doing
nothing
.
Input signals that are brief enough to have these three properties are
called
impulses

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