Solutions to HW Problems from Chapter 6
Problem 6.2
Use the graphical method to convolve
p
P
@
t
A
with
p
P
@
t
A
.
SOLUTION: The convolution duration property implies that
p
P
@
t
A
p
P
@
t
A a H
for
t
P a
I
I
and
t
P a I C I
.
For
P
t
H
, the convolution produces
p
P
@
t
A
p
P
@
t
A a
IC
t
I
I
d
a
t
C P
For
H
t
P
, the convolution produces
p
P
@
t
A
p
P
@
t
A a
I
IC
t
I
d
a P
t
Problem 6.4
Use the graphical method to convolve triangularly shaped signals presented in FIGURE 6.29.
f
1
(t)
f (t)
2
0
1
1
t
0
1
2
1
t
FIGURE 6.29
SOLUTION: By the convolution duration property, we have that
f
I
@
t
A
f
P
@
t
A a H
for
t
H C I a I
and
t
I C P a Q
.
In the time interval
I
t
P
, the convolution is given by
f
I
@
t
A
f
P
@
t
A a
t
I
H
@
I C
t
A
d
a
@
t
IA
Q
Q
C
@
t
IA
P
P
@
t
IA a
I
T
@
t
IA
Q
In the time interval
P
t
Q
, the convolution is given by
f
I
@
t
A
f
P
@
t
A a
I
t
P
@
I C
t
A
d
a
I
Q
C
I
P
@
t
IA C
@
t
PA
Q
Q
@
t
PA
P
P
@
t
IA
181
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View Full DocumentCHAPTER 6
Problem 6.5
Convolve graphically the signals given in FIGURE 6.30.
f
1
(t)
f (t)
2
t
0
1
2
1
0
1
1
t
FIGURE 6.30
SOLUTION: By the convolution duration property, we have that the convolution is equal to zero
for
t
H C H a H
and
t
I C P a Q
.
In the time interval
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 Spring '11
 Hoa
 Digital Signal Processing, LTI system theory, Impulse response, 0 k

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