EE 378
Handout #17
Statistical Signal Processing
Wednesday, May 30, 2007
Homework Set #8
Due: Wednesday, June 6, 2007.
Announcement:
You can hand in the HW either after class or deposit it, before 5pm,
in the
Homework in
box in the 378 drawer of the class file cabinet on the second floor of
the Packard Building.
1. Let
Y
be a an Nlength vector with independent elements (i.e.
Y
1
,...
Y
N
are indepen
dent). Assume
Y
i
, 1
≤
i
≤
N
, is Poisson with parameter
θ
.
(a) Compute the CramerRao bound for estimating
θ
based on
Y
.
(b) Find
ˆ
θ
ML
, the maximumlikelihood estimate of
θ
based on
Y
.
(c) Is
ˆ
θ
ML
unbiased?
(d) Is
ˆ
θ
ML
consistent?
(e) Is
ˆ
θ
ML
efficient?
Now, instead assume
Y
i
, 1
≤
i
≤
N
, is Laplacian with parameter
θ
, i.e.
f
(
y
i
) =
1
2
θ
e
−

y
i

θ
,
and that
ˆ
θ
= (1
/N
+
a/N
2
)
∑
N
i
=1

Y
i

.
(f) Is
ˆ
θ
unbiased?
(g) Find
MSE
(
ˆ
θ
).
(h) Find the value of
a
that minimizes
MSE
(
ˆ
θ
).
(i) Repeat parts (a) through (e) for the Laplacian case, with the given
ˆ
θ
and the
value
a
found in part (h).
(j) There is an efficient estimator
ˆ
θ
(
Y
) such that
ˆ
θ
(
Y
) =
θ
+
parenleftbigg
1
I(
θ
)
parenrightbigg parenleftbigg
∂
ln
f
θ
(
Y
)
∂θ
parenrightbigg
.
For both cases (the Poisson and the Laplacian), find the
ˆ
θ
(
Y
) using this formula.
Explain your results.
2. In order to estimate
N
0
/
2, the amplitude of the power spectral density of
W
(
t
), a zero
mean, white Gaussian noise process, we use
{
θ
i
(
t
)
,
0
≤
t
≤
T
}
,
1
≤
i
≤
n
, a family of
n
real, orthonormal functions, i.e.
integraldisplay
T
0
θ
i
(
t
)
θ
j
(
t
)
dt,
=
δ
ij
,
1
≤
i, j
≤
n.
1
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We calculate
Y
i
=
integraldisplay
T
0
W
(
t
)
θ
i
(
t
)
dt,
1
≤
i
≤
n,
and estimate
N
0
/
2 based on
Y
1
, ...Y
n
.
(a) Find the maximum likelihood estimate of
N
0
2
based on
Y
i
, 1
≤
i
≤
n
.
(b) Express the bias and the MSE of the estimator as a function of
N
0
and
n
.
(c) Compute the CramerRao bound. Determine whether the estimator is efficient.
(d) Define a function sequence
{
φ
i
(
t
)
,
0
≤
t
≤
T
}
, 1
≤
i
≤
n
, where
φ
i
(
t
) =
αφ
i
−
1
(
t
) +
θ
i
(
t
)
,
1
≤
i
≤
n,
where
θ
0
(
t
) = 0 and
α
is a real number.
Repeat parts (a) through (c) with
Y
replaced by
Z
, where
Z
i
=
integraldisplay
T
0
W
(
t
)
φ
i
(
t
)
dt,
1
≤
i
≤
n.
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 Spring '07
 Weissman,I
 Signal Processing, Maximum likelihood, Estimation theory, Yi, ML estimate, noisy image

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