107
CHAPTER
6
Convolution
Convolution is a mathematical way of combining two signals to form a third signal.
It is the
single most important technique in Digital Signal Processing.
Using the strategy of impulse
decomposition, systems are described by a signal called the
impulse response
.
Convolution is
important because it relates the three signals of interest: the input signal, the output signal, and
the impulse response.
This chapter presents convolution from two different viewpoints, called
the input side algorithm and the output side algorithm.
Convolution provides the mathematical
framework for DSP; there is nothing more important in this book.
The Delta Function and Impulse Response
The previous chapter describes how a signal can be decomposed into a group
of components called
impulses
.
An impulse is a signal composed of all zeros,
except a single nonzero point.
In effect, impulse decomposition provides a way
to analyze signals one sample at a time.
The previous chapter also presented
the fundamental concept of DSP:
the input signal is decomposed into simple
additive components, each of these components is passed through a linear
system, and the resulting output components are synthesized (added).
The
signal resulting from this divideandconquer procedure is identical to that
obtained by directly passing the original signal through the system.
While
many different decompositions are possible, two form the backbone of signal
processing: impulse decomposition and Fourier decomposition.
When impulse
decomposition is used, the procedure can be described by a mathematical
operation called
convolution
.
In this chapter (and most of the following ones)
we will only be dealing with
discrete
signals.
Convolution also applies to
continuous
signals, but the mathematics is more complicated. We will look at
how continious signals are processed in Chapter 13.
Figure 61 defines two important terms used in DSP.
The first is the
delta
function
, symbolized by the Greek letter delta,
.
The delta function is
*
[
n
]
a
normalized
impulse, that is, sample number zero has a value of one, while
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108
all other samples have a value of zero.
For this reason, the delta function is
frequently called the
unit impulse
.
The second term defined in Fig. 61 is the
impulse response
.
As the name
suggests, the impulse response is the signal that exits a system when a delta
function (unit impulse) is the input.
If two systems are different in any way,
they will have different impulse responses.
Just as the input and output signals
are often called
and
, the impulse response is usually given the
x
[
n
]
y
[
n
]
symbol,
.
Of course, this can be changed if a more descriptive name is
h
[
n
]
available, for instance,
might be used to identify the impulse response of
f
[
n
]
a
filter
.
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 Summer '09
 Digital Signal Processing, Signal Processing, Impulse response

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