Digital Signal Processing Reference
In-Depth Information
h
(
t
)
x
(
t
)
2.0
1.5
t
t
−1
0
2
−2
0
3
(a)
(b)
h
(−
t
)
h
(
t
−
t
)
2.0
2.0
t
t
0
1
(
t
− 2)
(
t
+1)
0
−2
(c)
(d)
x
(
t
),
h
(
t
−
t
)
case 1:
t
< −3
x
(
t
),
h
(
t
−
t
)
case 2: −3 ≤
t
< 0
2.0
2.0
1.5
1.5
t
t
(
t
− 2)
(
t
+1)
−2
0
3
3
(
t
− 2)
−2
0
(
t
+1)
(e)
(f)
x
(
t
),
h
(
t
−
t
)
case 3: 0 ≤
t
<1
case 4: 2 ≤
t
< 5
x
(
t
),
h
(
t
−
t
)
2.0
2.0
1.5
1.5
t
t
−2
0
(
t
− 2)
3
(
t
+1)
(
t
+1)
−2
(
t
− 2)
(g)
(h)
x
(
t
),
h
(
t
−
t
)
case 5:
t
≥ 5
2.0
Fig. 3.10. Convolution of the
input signal
x
(
t
) with the
impulse response
h
(
t
)in
Example 3.9. Parts (a)-(i) are
discussed in the text.
1.5
t
−2
0
3
(
t
− 2)
(
t
+1)
(i)
Combining the five cases, we obtain
0
t
< −
3
3(
t
+
3)
−
3
≤
t
≤
0
y
(
t
)
=
≤
t
≤
2
3(5
−
t
)
2
≤
t
≤
5
0
t
>
5
.
The waveform for the output response is sketched in Fig. 3.11.
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