EE538
Final Exam
Fall 2007
Mon, Dec 10, 810 am
RHPH 127
Dec. 10, 2007
Cover Sheet
Test Duration: 120 minutes.
Open Book but Closed Notes.
Calculators allowed!!
This test contains
five
problems.
Each of the
five
problems are equally weighted.
All work should be done in the blue books provided.
You must show all work for each problem to receive full credit.
Do
not
return this test sheet, just return the blue books.
1
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EE538 Digital Signal Processing I
Final Exam
Fall 2007
Problem 1.
This problem deals with the properties of deterministic autocorrelation.
(a) Consider the symmetric sequence below which is one for
−
2
≤
m
≤
2 and zero for

m

>
2. Is this is a valid autocorrelation sequence? Justify your answer.
r
xx
[
m
] =
u
[
m
+ 2]
−
u
[
m
−
3]
(b) Consider the symmetric sequence below. Is this is a valid autocorrelation sequence?
You need to explain your answer.
r
xx
[
m
] = (3
− 
m

)(
u
[
m
+ 2]
−
u
[
m
−
3])
(c) Let
r
xx
[
m
] denote the autocorrelation sequence for the DT signal
x
[
n
].
Let
y
[
n
] =
x
[
n
−
n
o
], where
n
o
is an integer. Let
r
yy
[
m
] denote the autocorrelation sequence for
the DT signal
y
[
n
]. Derive an expression relating
r
yy
[
m
] and
r
xx
[
m
]. That is, how is
r
yy
[
m
] related to
r
xx
[
m
]?
(d) Let
r
xx
[
m
] denote the autocorrelation sequence for the DT signal
x
[
n
].
Let
y
[
n
] =
e
j
(
ω
0
n
+
θ
)
x
[
n
], where
ω
o
is some frequency and
θ
is some phase value. Let
r
yy
[
m
] denote
the autocorrelation sequence for the DT signal
y
[
n
].
Derive an expression relating
r
yy
[
m
] and
r
xx
[
m
]. That is, how is
r
yy
[
m
] related to
r
xx
[
m
]?
(e) Consider that the signal
x
[
n
] below, where
a
=
1
2
, is the input to the LTI system
described by the difference equation in equation (2) below.
x
[
n
] =
a
n
u
[
n
]
−
1
a
a
n
−
1
u
[
n
−
1]
(1)
y
[
n
] =
1
4
y
[
n
−
1] +
x
[
n
]
(2)
(i) Determine a closedform analytical expression for the autocorrelation
r
xx
[
m
] for
x
[
n
], when
a
=
1
2
. (Hint: examine the ZTransform of
x
[
n
].)
(ii) Determine a closedform analytical expression for the crosscorrelation
r
yx
[
m
] be
tween the input and output.
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 Fall '08
 ZOLTOLSKI
 Digital Signal Processing, Signal Processing, Autocorrelation, Stationary process, autocorrelation sequence, a1 e−jω 2

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