Image Processing Reference
In-Depth Information
discussion, we will elaborate on the estimation of the underlying signal amplitude
from a set of data points of which this signal amplitude is assumed to be the
same (i.e., a constant model). It is, however, clear that similar reasoning is valid
for any other underlying (parametric) model of the data points.
For both data sets (complex and magnitude), the ML estimators of the signal
amplitude will be derived. The use of the ML estimator is justified by the fact that
the ML estimator has a number of favorable statistical properties, as discussed in
Section 4.3.6 [25]. First, it is asymptotically precise, i.e., it achieves the so-called
CRLB for an infinite number of observations. The CRLB defines a lower bound
on the variance of any unbiased estimator of a parameter. Second, the ML estimator
is consistent, which means that it converges to the true parameter in a statistically
well-defined way if the number of observations (i.e., data points) increases. Third,
the ML estimator is asymptotically normally (i.e., Gaussian) distributed, with a
mean equal to the true value of the parameter(s) and a (co)variance (matrix) equal
to the CRLB. Whether these asymptotic properties also apply when the number of
observations is finite depends on the particular estimation problem under concern.
In the present case, this can be found out analytically (for complex data points) or
by means of simulations (for magnitude data points). It is known that if there exists
an estimator that attains the CRLB, it is given by the ML estimator [25]. For both
data sets, the performance of the corresponding ML estimators of the signal ampli-
tude will be evaluated in terms of the MSE, a measure of both accuracy (bias) and
precision (variance). Moreover, for both complex and magnitude data, the variance
of the ML estimator will be compared with the CRLB, which can be computed
analytically. In addition, for both types of data sets, the ML estimators of the
variance of the noise will be derived, after which their performance will be evaluated
in terms of both accuracy and precision [37].
4.4.2
S
IGNAL
A
MPLITUDE
E
STIMATION
FROM
C
OMPLEX
D
ATA
We start by considering complex, Gaussian-distributed data. The CRLB for unbi-
ased estimation of the underlying amplitude signal as well as the ML estimator
of this signal will be derived. This will be done for data with identical underlying
phase values, as well as for data with different underlying phase values.
4.4.2.1
Region of Constant Amplitude and Phase
Consider a set of
N
independent, Gaussian-distributed, complex data points
with underlying true amplitude and phase values
A
and
c
=
{(
ww
rn
,
)}
ϕ
, respec-
,
in
,
tively. This means that
A
cos
represent the true real and imaginary
values, respectively. As the real and imaginary data are independent, the joint
PDF of the complex data, is simply the product of the marginal PDFs of the
Gaussian-distributed real and imaginary data points:
ϕ
and
A
sin
ϕ
p
c
,
N
N
=
2
2
(
ω
rrn
A
,
−
cos
ϕ
)
(
ω
in
A
,
−
sin
ϕ
)
1
∏
−
−
p
({(
,
)}
| ,
A
ϕ
)
e
e
,
(4.53)
ww
2
2
2
σ
2
σ
c
rn
,
in
,
2
πσ
2
n
=
1
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