EE 562a
Homework Set 8
Due Wednesday 25 April 2007
1
(The following welldefined problems come from different sources, and the notation used may vary.
Don’t let that bother you!)
1. Consider an LTI system,
M
, characterized by the following differential equation:
˜
y
=
M
˜
x
⇐⇒
¨
y
(
t
) + 3 ˙
y
(
t
) + 2
y
(
t
) = 3
x
(
t
)

˙
x
(
t
)
.
(1)
A continuous time, wss random process,
x
(
u, t
), with correlation function
R
x
(
τ
) =
1
8
e

4

τ

is passed through the above LTI system, with the output denoted by
y
(
u, t
).
(a) What is the frequency response of the system  i.e.
M
(
f
)?
(b) Determine
S
y
(
f
) and
S
xy
(
f
).
(c) Determine the optimal (Wiener) causal filter for estimating
x
(
u, t
) from
y
(
u, t
); specify
the frequency response of this filter.
(d) What is the PSD of the best estimate, ˆ
x
(
u, t
), in terms of
S
x
(
f
)?
Hint:
Determine the frequency response of the cascade of
M
(
f
) and the Wiener filter
found in the previous part.
(e) What is the associated MMSE of this estimator?
(f) Explain how your solution would change if the righthand side of (1) were 3
x
(
t
) + ˙
x
(
t
).
What is the system characteristic which is changed by this sign change.
2. Let
w
(
u, t
),
x
(
u, t
),
y
(
u, t
), and
z
(
u, t
), be zeromean widesense stationary random processes.
(a) Supply the correlation function or power spectral density, which ever is not given. Here
α
and
σ
are positive constants.
i.
R
w
(
τ
) =
σ
2
e

ατ
2
,
∀
τ
∈
R
.
ii.
R
x
(
τ
) =
σ
2
sin 2
τ
τ
,
∀
τ
∈
R
.
iii.
R
y
(
τ
) =
σ
2
1 +
ατ
2
exp(
i
60
πτ
)
,
∀
τ
∈
R
.
iv.
S
z
(
f
) =
sin
2
(
πfT
)
T
(
πf
)
2
,
∀
f
∈
R
.
(b) Which of these four processes have a power spectral densities that are factorable in the
form of a causal system function times its conjugate?
(c) Which of these four processes are differentiable in the mean square sense?
(d) Which of these four processes are meanergodic?
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EE 562a
Homework Set 8
Due Wednesday 25 April 2007
3.
A clock signal detector
(Final Exam, Spring 1990)
The following circuit is designed to produce a periodic signal with period
T
from a data signal
x
(
u, t
) of the form
x
(
u, t
) =
∞
n
=
∞
a
n
(
u
)
p
(
t

nT
)
where
{
a
n
(
u
)
}
is a sequence of independent identically distributed random variables, equally
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 Spring '07
 ToddBrun
 Signal Processing, Probability theory, LTI system theory, power spectral density, Bvid

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