•
2. Let Y=X+N, where X and N are independent zero-
mean Gaussian rv’s with different variances.
–
a. Find the mmse linear estimator for Y
2
2
2
2
2
2
2
[
]
[
]
[
]
0
[
]
[
]
[
]
[
]
(
,
)
[
]
(
)
X
N
X
X
X
X
Y
X
Y
X
Y
X
X
N
E Y
E X
E N
Var Y
Var X
Var N
E XY
COV X Y
E XY
σ
σ
σ
σ
σ
ρ
σ σ
σ σ
σ σ
σ
σ
σ
=
+
=
=
+
=
+
=
=
=
=
=
+
,
2
ˆ
(
[
])
[
]
Y
X Y
X
X
Y
X
Y
X
Y
X
E X
E Y
X
X
σ
ρ
σ
σ
σ
σ σ
σ
=
−
+
=
=
mmse linear
estimator:
2
2
2
[(
) ]
[
]
N
mse
E Y
X
E N
σ
=
−
=
=

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