Digital Signal Processing Reference
In-Depth Information
Next, the
k
-axis is plotted below the
m
-axis in Figure 10.4(a), resulting in the plot for
in Figure 10.4(b). The second factor in convolution summation,
h
[
k
], was calculated
in Example 10.1 and is also shown in Figure 10.4(a).
First, the two factors of the convolution summations, and
h
[
k
], are plotted in
Figure 10.4(c) for We see that for each nonzero value in one of the signals, the other
signal has a value in zero. Hence, the product
x[n - k]
x[n - k]
n = 0.
x[n - k]h[k]
is zero for
n = 0
and all
k
, and it
follows that
y[0] = 0.
Furthermore, we see that
y[n] = 0
for all
n 6 0,
since the plot of
x[n - k]
is shifted to the left for
Figure 10.4(d) gives
n 6 0.
x[n - k]
and
h
[
k
] for
n = 1.
For this value of
n
,
x[n - k]h[k] ƒ
n= 1, k= 0
= x[1]h[0] = (1
>
3)(3) = 1,
and this product is zero for all other
k
. Thus,
Figure 10.4(e) shows
y[1] = 1.
x[n - k]
and
h
[
k
] for
n = 2.
All product terms of the convolution
sum are zero, except for
k = 0
and
k = 1.
Then
y[2] = x[2]h[0] + x[1]h[1] = 1.5 + 1 = 2.5.
Using the same procedure, we see that
y[3] = 4.5, y[4] = 3.5, y[5] = 2,
and
y[n] = 0,
for
n 7 5.
The output is then
y[n] = d[n - 1] + 2.5d[n - 2] + 4.5d[n - 3]
+ 3.5d[n - 4] + 2d[n - 5].
The output signal is plotted in Figure 10.4(f).
The output can also be calculated from the expansion of the convolution sum:
y[n]
Á
y[n] =
+ x[n - 3]h[3] + x[n - 2]h[2]
Á
+ x[n - 1]h[1] + x[n]h[0] + x[n + 1]h[-1] +
= x[n - 2]h[2] + x[n - 1]h[1] + x[n]h[0].
This expansion has only three nonzero terms, because
h
[
n
] is nonzero for only As
a third method of evaluation, the input signal can be traced through the system in Figure 10.3,
as suggested in Example 10.1. Finally, the results can be verified with the MATLAB program
n = 0, 1, 2.
n=1:5;
x=[3 4.5 6];
h=[1/3 1/3 1/3];
y=conv(x,h)
stem(n,y,'fill'), grid
■
Note that in Examples 10.1 and 10.2, three different descriptions of the system
are given:
1.
the impulse response in (10.20);
2.
the difference equation in (10.19);
3.
a block diagram in Figure 10.3.
As a practical matter, in technical literature, both the difference equation and a
block diagram are usually given as the description of a discrete-time system. The
