123
CHAPTER
7
EQUATION 71
The delta function is the identity for
convolution. Any signal convolved with
a delta function is left unchanged.
x
[
n
]
(
*
[
n
]
’
x
[
n
]
Properties of Convolution
A linear system's characteristics are completely specified by the system's impulse response, as
governed by the mathematics of convolution.
This is the basis of many signal processing
techniques.
For example:
Digital filters are created by
designing
an appropriate impulse
response.
Enemy aircraft are detected with radar by
analyzing
a measured impulse response.
Echo suppression in long distance telephone calls is accomplished by creating an impulse
response that
counteracts
the impulse response of the reverberation.
The list goes on and on.
This chapter expands on the properties and usage of convolution in several areas.
First, several
common impulse responses are discussed.
Second, methods are presented for dealing with
cascade and parallel combinations of linear systems.
Third, the technique of
correlation
is
introduced.
Fourth, a nasty problem with convolution is examined, the computation time can be
unacceptably long using conventional algorithms and computers
.
Common Impulse Responses
Delta Function
The simplest impulse response is nothing more that a delta function, as shown
in Fig. 71a.
That is, an impulse on the input produces an identical impulse on
the output.
This means that
all
signals are passed through the system
without
change
.
Convolving any signal with a delta function results in exactly the
same signal.
Mathematically, this is written:
This property makes the delta function the
identity
for convolution.
This is
analogous to
zero
being the identity for addition
, and
one
being the
(
a
%
0
’
a
)
identity for multiplication
.
At first glance, this type of system
(
a
×1
’
a
)
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The Scientist and Engineer's Guide to Digital Signal Processing
124
EQUATION 72
A system that amplifies or attenuates has
a scaled delta function for an impulse
response.
In this equation,
k
determines
the amplification or attenuation.
x
[
n
]
(
k
*
[
n
]
’
kx
[
n
]
EQUATION 73
A relative shift between the input and
output signals corresponds to an impulse
response that is a shifted delta function.
The variable, s, determines the amount of
shift in this equation.
x
[
n
]
(
*
[
n
%
s
]
’
x
[
n
%
s
]
may seem trivial and uninteresting.
Not so!
Such systems are the ideal for
data storage, communication and measurement.
Much of DSP is concerned
with passing information through systems without change or degradation.
Figure 71b shows a slight modification to the delta function impulse
response.
If the delta function is made larger or smaller in amplitude, the
resulting system is an
amplifier
or
attenuator
, respectively.
In equation
form, amplification results if
k
is
greater than one
, and attenuation results
if
k
is
less than one
:
The impulse response in Fig. 71c is a delta function with a
shift
.
This results
in a system that introduces an identical shift between the input and output
signals.
This could be described as a signal
delay
, or a signal
advance
,
depending on the direction of the shift.
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 Summer '09
 Digital Signal Processing, Signal Processing, Impulse response, sample number

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