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?