Image Processing Reference
In-Depth Information
where
σ
denotes the standard deviation of the noise, and
{(
ωω
rn
,
{(
)}
are the real
,
in
,
and imaginary variables corresponding with the complex data
ww
,
,
)}.
rn
in
,
4.4.2.1.1 CRLB
It follows from Subsection 4.3.5 that the CRLB for unbiased estimation of (
A
,
ϕ
)
can be computed from the Fisher information matrix
I
[25]:
∂
ln
p
∂
∂∂
ln
p
2
2
N
c
c
0
∂
A
2
A
ϕ
σ
2
I
=−
=
(4.54)
E
∂
∂∂
2
ln
p
A
∂
2
ln
p
NA
2
c
c
0
σ
2
ϕ
∂
ϕ
2
with the joint PDF
p
c
given by Equation 4.42. Applying the inverse operator
yields for the CRLB:
σ
2
0
N
CRLB
==
I
−
1
(4.55)
σ
2
0
NA
2
4.4.2.1.2 ML Estimation
Following the procedure described in Subsection 4.3.6, the likelihood function
L
is
obtained by substituting the available observations
{(
ww
rn
,
)}
for
{(
ωω
rn
,
)}
in
,
in
,
,
in
,
the joint PDF (4.53):
N
N
2
2
=
1
(
w
r
,,
−
A
cos
ϕ
)
(
w
in
A
,
−
sin
ϕ
)
∏
n
−
−
LA
(
,|
ϕ
{(
w
,
w
)})
e
e
.
(4.56)
2
2
2
σ
2
σ
rn
,
in
,
2
πσ
2
n
=
1
Then, the ML estimates of (
A
,
ϕ
) are found by maximizing this function with
respect to
A
and
ϕ
. Taking the logarithm yields:
N
1
∑
ln
LN
=−
ln(
2
πσ
2
)
+
[(
w A
−
cos
ϕ
)
2
+
(
w
−
A
sin
ϕ
) ].
2
(4.57)
rn
,
i
,
n
2
σ
2
n
=
1
At the maximum, the first derivative of ln
L
with respect to
A
and
ϕ
should
be zero:
N
∂
ln
L
A
NA
1
∑
=−
(
w
s
ϕ
+
w
sin)
ϕ
,
(4.58)
rn
,
in
,
∂
σσ
2
2
n
=
1
N
∂
ln
LA
∑
=
(
w
sin
ϕ
−
w
s
ϕ
).
(4.59)
rn
,
in
,
∂
ϕσ
2
n
=
1
Search WWH ::
Custom Search