Image Processing Reference
In-Depth Information
Hence, the CRLB for unbiased estimation of (
A
,
ϕ
,
σ
2
) is given by
σ
2
0
0
N
σ
2
CRLB
==
I
−
1
0
0
.
(4.135)
NA
2
σ
4
0
0
N
4.5.2.1.2 ML Estimation
For identical true phase values, the ML estimator of
σ
2
can be shown to be given by
N
1
∑
2
2
(
ˆ
(
ˆ
ˆ
cos
ˆ
ˆ
σ
2
=
A
ϕ
−
)
+
A
sin
ϕ
ML
−
)
,
(4.136)
w
w
ML
2
N
ML
ML
rn
,
ML
in
,
n
=
1
A
ϕ
with
and
given by Equation 4.60 and Equation 4.61, respectively.
ML
ML
2
ˆ
Notice that for
N
=
1, the estimator
σ
ML
will be equal to zero.
4.5.2.1.3 MSE
It can be shown that for large
N
the quantity
2
N
ˆ
2
σσ
ML
/
2
is approximately distrib-
uted as
2 degrees of freedom).
Because the mean and variance of a chi-squared variable with
χ
22
2
N
(i.e., chi-square distributed with 2
N
−
−
λ
degrees of
freedom are given by
λ
and 2
λ
, respectively, we find for the bias and variance
2
ˆ
of
σ
ML
that
( )
σ
2
σ
2
2
ˆ
b
σ
(
22
N
−− =−
)
σ
2
(4.137)
ML
2
N
N
and
( )
σ
4
1
2
ˆ
Var
σ
NN
1
−
,
(4.138)
ML
2
ˆ
respectively. Then, the MSE of
σ
ML
is given by
( )
N
σ
4
2
ˆ
MSE
σ
.
(4.139)
ML
4.5.2.2
Region of Constant Amplitude and Different Phases
Next, consider a region with a constant nonzero underlying signal amplitude and
different underlying phase values.
Search WWH ::
Custom Search