1
EE113 Homework 5
Problem 6.7
Evaluate the convolution sum
u
(

n
+2)
⋆
parenleftbigg
1
3
parenrightbigg
n

1
·
u
(
n

2)
using both the analytical and graphical methods. Compare your results.
Problem 6.9
Evaluate
parenleftbigg
1
2
parenrightbigg
n

1
·
u
(
n
)
⋆
bracketleftbiggparenleftbigg
1
3
parenrightbigg
n
·
u
(
n

2)

u
(
n
+1)
bracketrightbigg
using the distributivity property of the convolution sum.
Problem 6.23
Consider two possibly complexvalued sequences
x
(
n
)
and
h
(
n
)
. Their crosscorrelation is the
sequence
r
xh
(
n
)
whose samples are defined as follows:
r
xh
(
n
)=
∞
summationdisplay
k
=
∞
x
(
k
)
h
*
(
k

n
)
(a) Verify that
r
xh
(
n
)=
x
(
n
)
⋆h
*
(

n
)
.
(b) Use the graphical method to evaluate the correlation of the two sequences:
x
(
n
)=
{
2
,
1
,

1
,
2
}
and
h
(
n
)=
{
0
,
1
,
2
}
Problem 6.31
A causal LTI system is described by the difference equation
y
(
n
)=
1
4
y
(
n

1)+
x
(
n

1)
Find its impulse response sequence. Find also the response to the input sequence shown in Fig. 6.18 using the
convolution sum method.
Problem 7.9
Determine the solution of each of the following homogeneous equations with initial conditions:
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 Fall '08
 Walker
 Digital Signal Processing, Convolution sum, LTI system theory

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