Digital Signal Processing Reference
In-Depth Information
x
[
n
]
y
1
[
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]
y
[
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]
x
[
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]
y
[
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]
h
1
[
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]
h
2
[
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]
h
2
[
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]
h
1
[
n
]
(a)
x
[
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]
y
[
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]
x
[
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]
y
[
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h
1
[
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h
2
[
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h
1
[
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h
2
[
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]
Figure 10.8
Associative property.
(b)
As an example of this property, consider the output of the system of
Figure 10.8(a), which is given by
y[n] = y
1
[n]
*
h
2
[n] = (x[n]
*
h
1
[n])
*
h
2
[n].
Then, by Property 2,
(x[n]
*
h
1
[n])
*
h
2
[n] = x[n]
*
(h
1
[n]
*
h
2
[n]) = x[n]
*
(h
2
[n]
*
h
1
[n]).
(10.25)
Hence, the order of the two systems of Figure 10.8(a) may be reversed without
changing the input-output characteristics of the total system, as shown in Figure
10.8(a).
Also, from (10.25), the two cascaded systems of Figure 10.8(a) may be re-
placed with a single system with the impulse response
h[n] = h
1
[n]
*
h
2
[n],
(10.26)
such that the input-output characteristics are preserved. This property is illustrated
in Figure 10.8(b). It follows that for
m
cascaded LTI systems, the impulse response
of the total system is given by
h[n] = h
1
[n]
*
h
2
[n]
*
Á
*
h
m
(n).
3.
Distributive property
. The convolution sum satisfies the following rela-
tionship:
x[n]
*
h
1
[n] + x[n]
*
h
2
[n] = x[n]
*
(h
1
[n] + h
2
[n]).
(10.27)
We prove this relation by using the convolution sum of (10.13):
x[n]
*
h
1
[n] + x[n]
*
h
2
[n] =
q
k=-
q
x[k]h
1
[n - k] +
q
k=-
q
x[k]h
2
[n - k]
=
q
k=-
q
x[k](h
1
[n - k] + h
2
[n - k])
= x[n]
*
(h
1
[n] + h
2
[n]).
