Digital Signal Processing Reference
In-Depth Information
x
( )
1
0
(a)
h
(
t
)
1
t
0
(b)
h
(
t
)
1
0
t
(c)
x
( )
h
(
t
)
1
0
t
(d)
Figure 3.6
Convolution for Example 3.4.
The -axis is plotted in Figure 3.5(a) and is the same as the axis of
t
h(t - t)
in Fig-
ure 3.5(c).
Shown in Figure 3.6(a) is the first term of the convolution integral,
Figure 3.6(b) shows the second term of the integral, for The product of these
two functions is zero; hence, the value of the integral [and of ] is zero for Fig-
ure 3.6(c) shows the second term of the convolution integral for and Figure 3.6(d)
shows the product of the functions, of Figure 3.6(a) and (c). Therefore, from
the convolution integral, is the area under the function in Figure 3.6(d). Because the
product function is triangular, the area is equal to one-half the base times the height:
x(t) = tu(t).
h(t - t),
t 6 0.
y(t)
t 6 0.
t 7 0,
x(t)h(t - t),
y(t)
t
2
2
,
1
2
(t)(t) =
y(t) =
t 7 0.
This value is the same as that found in Example 3.2.
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