EE 200 – Fall 2009 (Weber)
Homework 4
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Date
EE 200, Homework #
Show intermediate steps whenever possible.
1.
Three systems are connected in cascade: the output of
S
1
goes to the input of
S
2
, the output of
S
2
is
the input to
S
3
. The systems are specified as follows.
S
1
:
y
1
[
n
] =
x
1
[
n
]

x
1
[
n

1]
S
2
:
y
2
[
n
] =
x
2
[
n
] +
x
2
[
n

2]
S
3
:
y
3
[
n
] =
x
3
[
n

1] +
x
3
[
n

2]
a.
What is the frequency response of the overall system,
S
, with input
x
[
n
] =
x
1
[
n
] and output
y
[
n
] =
y
3
[
n
]?
b.
Based on your result in part a, Write a difference equation similar to the ones above that describes
the system
S
in terms of
x
[
n
] and
y
[
n
].
2.
A system has an impulse response of
h
[
n
] =
δ
[
n
]

δ
[
n

3].
a.
What is the frequency response of the system?
b.
Determine the output of the system when the input is
x
[
n
] = cos(2
πn/
3)
c.
Determine the output of the system when the input is
x
[
n
] = cos(
πn/
3 +
π/
2).
d.
Determine the output of the system when the input is
x
[
n
] = cos(
πn/
6 +
π/
2).
3.
For each of the following frequency responses, determine the impulse response
h
[
n
].
a.
H
(
e
j
ˆ
ω
) = 3 + 4
e

j
3ˆ
ω
b.
H
(
e
j
ˆ
ω
) = 5
e

j
3ˆ
ω
cos(ˆ
ω
)
c.
H
(
e
j
ˆ
ω
) =
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 '08
 ZADEH
 Digital Signal Processing, LTI system theory, Impulse response

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