Digital Signal Processing Reference
In-Depth Information
Section 9.4.) The results of this example can be verified with the following MATLAB program:
% This MATLAB program performs the convolution operation of
% Example 10.4.
% Establish the input unit-step function.
x=ones (1,11);
% Compute the system impulse response.
for k = 1:10;
h(k) = 0.6^(k-1);
end
% Convolve the input with the impulse response and plot.
c=conv (x,h);
for k=1:11;
n(k)=k-1;
y(k)=c(k);
end
[n' y']
stem(n, y, 'fill')
■
We now discuss three properties of the convolution sum that are related to systems:
1.
Commutative property
. The convolution sum is symmetric with respect to
x
[
n
] and
h
[
n
]:
x[n]
*
h[n] = h[n]
*
x[n].
(10.23)
This property was derived in (10.16) and is illustrated in Figure 10.7. In this figure,
an LTI system is represented by a block containing the impulse response. The out-
put for each system is identical, from (10.23).
2.
Associative property
. In the convolution of three signals, the result is
the same, independent of the order that the convolution is performed. For
example,
(f[n]
*
g[n])
*
h[n] = f[n]
*
(g[n]
*
h[n]) = (h[n]
*
f[n])
*
g[n].
(10.24)
The proof of this property is not given. (See Problem 10.13.)
x
[
n
]
y
[
n
]
h
[
n
]
y
[
n
]
h
[
n
]
x
[
n
]
Figure 10.7
Commutative property.
