EE 5375/7375 Random Processes
October 23, 2003
Homework #7 Solutions
Problem 1. textbook problem 7.2
Let
p
(
x
) be the rectangular function shown in Fig. P7.2. Is
R
X
(
τ
) =
p
(
τ/T
) a valid autocorrelation function?
R
X
(
τ
) is a rectangle with height
A
in the interval (

T, T
).
According to Appendix B in the textbook,
its Fourier transform is
A
2
T
sin(2
πfT
)
2
πfT
. Since this is negative sometimes, but power spectral densities must
always be nonnegative,
p
(
τ/T
) cannot be a valid autocorrelation function.
Problem 2. textbook problem 7.5
Let
Z
(
t
) =
X
(
t
) +
Y
(
t
). Under what conditions does
S
Z
(
f
) =
S
X
(
f
) +
S
Y
(
f
)?
We know (from notes and Example 7.4 in the textbook) that
R
Z
(
τ
) =
R
X
(
τ
) +
R
Y,X
(
τ
) +
R
X,Y
(
τ
) +
R
Y
(
τ
)
and
S
Z
(
f
) =
S
X
(
f
) +
S
Y,X
(
f
) +
S
X,Y
(
f
) +
S
Y
(
f
)
It is clear that
S
Z
(
f
) =
S
X
(
f
) +
S
Y
(
f
) only if
R
Y,X
(
τ
) +
R
X,Y
(
τ
) = 0. This would be true, for instance, if
R
Y,X
(
τ
) =
R
X,Y
(
τ
) = 0 for all
τ
; then
X
and
Y
are said to be orthogonal to each other.
Problem 3. textbook problem 7.6
Show that (a)
R
X,Y
(
τ
) =
R
Y,X
(

τ
) (b)
S
X,Y
(
f
) =
S
*
Y,X
(
f
).
(a)
R
X,Y
(
τ
) =
E
(
X
(
t
+
τ
)
Y
(
t
)) =
E
(
Y
(
t
)
X
(
t
+
τ
)) =
R
Y,X
(

τ
)
(b) The spectral density is the Fourier transform of the autocorrrelation function:
S
X,Y
(
f
) =
∞
∞
R
X,Y
(
τ
)
e

j
2
πfτ
dτ
=
∞
∞
R
Y,X
(

τ
)
e

j
2
πfτ
dτ
=
∞
∞
R
Y,X
(
τ
)
e
j
2
πfτ
dτ
=
S
*
Y,X
(
f
)
The last step follows from observing that the usual Fourier transform uses
e

j
2
πfτ
= cos(2
πfτ
)

j
sin(2
πfτ
)
in the integral, but the second to last step has
e
j
2
πfτ
= cos(2
πfτ
) +
j
sin(2
πfτ
). Therefore the result will
be the complex conjugate of the usual Fourier transform.
Problem 4. textbook problem 7.7
Let
Y
(
t
) =
X
(
t
)

X
(
t

d
). (a) Find
R
X,Y
(
τ
) and
S
X,Y
(
f
). (b) Find
R
Y
(
τ
) and
S
Y
(
f
).
(a) We find
R
X,Y
(
τ
) =
E
(
X
(
t
+
τ
)(
X
(
t
)

X
(
t

d
)))
=
E
(
X
(
t
+
τ
)
X
(
t
))

E
(
X
(
t
+
τ
)
X
(
t

d
))
=
R
X
(
τ
)

R
X
(
τ
+
d
)
Taking the Fourier transform,
S
X,Y
(
f
) =
S
X
(
f
)

S
X
(
f
)
e
j
2
πfd
1
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where we have used the property that a time shift of
T
in the time domain corresponds to an additional
exponential factor in the frequency domain, for example, a Fourier transform pair is
R
(
τ

T
)
⇔
S
(
f
)
e

j
2
πfT
(b) Similarly, we find
R
Y
(
τ
) =
E
((
X
(
t
+
τ
)

X
(
t
+
τ

d
))(
X
(
t
)

X
(
t

d
)))
= 2
R
X
(
τ
)

R
X
(
τ
+
d
)

R
X
(
τ

d
)
Taking the Fourier transform,
S
Y
(
f
) = 2
S
X
(
f
)

S
X
(
f
)
e
j
2
πfd

S
X
(
f
)
e

j
2
πfd
= 2
S
X
(
f
)(1

cos(2
πfd
))
Problem 5. textbook problem 7.8
Let
X
(
t
) and
Y
(
t
) be independent widesense stationary random processes, and define
Z
(
t
) =
X
(
t
)
Y
(
t
). (a)
Show that
Z
(
t
) is widesense stationary. (b) Find
R
Z
(
τ
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 Fall '08
 LORENZELLI
 inverse Fourier transform

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