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ELEC 531: STATISTICAL SIGNAL PROCESSING
Department of Electrical and Computer Engineering
Rice University
Fall 2007
Problem Set 4
Due: September 27, 2008
Problem 4.1
Let the observations be of the form
Y
=
H
θ
+
N
where
θ
and
N
are
statistically independent Gaussian random vectors.
θ
∼ N
(
0
,K
θ
)
N
∼ N
(
0
, K
n
)
The vector
θ
has dimension
M
; the vectors
Y
and
N
have dimension
L
.
(a) Derive
b
θ
MMSE
, the minimum meansquared error estimator of
θ
.
(b) Show that this estimator and the optimum linear estimator
b
θ
LMMSE
are equal.
(c) Find an expression for the meansquared error for these estimators.
Problem 4.2
Consider the communication system shown below. The message
X
is a
N
(0
,σ
2
X
) random variable. The transmitter output is
hX
, and the receiver input is
Y
=
hX
+
N,
where
N
in an
N
(0
,σ
2
N
) random variable that is statistically independent of
Y
.
RCVR
XMTR
+
X
N
hX
Y
X
Figure 1:
Suppose the transmitter is subject to intermittent failure, i.e.
h
is a random variable
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 Fall '07
 Lexa
 Signal Processing

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