Homework 4
ECE 220
Due: March 29, 2005
1. A LTI system is described the the difference equation
y
[
n
] =
1
3
(
x
[
n
] +
x
[
n

1] +
x
[
n

2])
(a) Determine the system function
H
(
z
) for this system.
(b) Plot the poles and zeros of
H
(
z
) in the zplane.
(c) From
H
(
z
) obtain an expression for
H
(
e
j
ˆ
ω
), the frequency response of the system.
(d) Sketch the frequency response (magnitude and phase) as a function of frequency for

π
≤
ˆ
ω
≤
π
.
(e) What is the output of the system if the input is
x
[
n
] = 2 + cos(
π
4
n
)

2 cos(
π
3
n
)
2. Figure 1 depicts a cascade connection of two LTI systems as described in Section 75.1 of your text.
(a) Use
z
transforms to show that the system function for the overall system (from
x
[
n
] to
y
[
n
]) is
H
(
z
) =
H
1
(
z
)
H
2
(
z
), where
Y
(
z
) =
H
(
z
)
X
(
z
).
(b) Use the result of (2a) to show that the order of the system is not important. That is, show that
for the same input
x
[
n
] to the systems, the overall outputs are the same so that
w
[
n
] =
y
[
n
].
(c) Suppose that both systems above are 3point averages described by the difference equation in
Problem 1. Determine the system function
H
(
z
) =
H
1
(
z
)
H
2
(
z
) for the overall system.
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 Spring '05
 JOHNSON
 Digital Signal Processing, LTI system theory

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