Image Processing Reference
In-Depth Information
Stated in another way, complex data that only consist of noise “point in all
directions” with the same probability.
•
For high SNR, it is easy to see that the probability of observing large
values for the phase deviation will be small. In that case, Equation 4.37
reduces to
1
2
A
−
∆
φ
σ
22
A
p
∆
∆=
φ
φ
()
p
.
(4.39)
π
σ
2
2
Thus, the phase noise
∆φ
is governed by a Gaussian distribution when
SNR
.
The standard deviations of the phase noise can in general be calculated from
Equation 4.37. However, for the SNR limits given in Equation 4.38 and Equation
4.39, it is given by
→
∞
!
!
π
/
3
if SNR
=
0
σ
∆
=
(4.40)
σ
/
A
if SNR
>>
1
.
4.3
PARAMETER ESTIMATION
4.3.1
P
ERFORMANCE
M
EASURES
OF
E
STIMATORS
In the remainder of this chapter, several estimators will be considered. In order
to compare the performance of these estimators, we require appropriate perfor-
mance measures. In this subsection, we will introduce two of the most commonly
used and widely accepted ones, namely
precision
and
accuracy
. Furthermore, the
mean squared error
(MSE), a measure incorporating both precision and accuracy,
will be introduced.
In what follows,
…
θθθθ
=, ,
(
)
T
represents an estimator of the
K
×
1 param-
1
K
eter vector
=
(
1
,
…
,
K
)
T
.
4.3.2
P
RECISION
The precision of an estimator concerns the spread of the estimates when the
experiment is repeated under the same conditions. It is represented by the variance
of the estimator or, equivalently, by the standard deviation of the estimator, which
is the square root of the variance. The variance is thus a measure of the nonsys-
tematic error. If the estimator is vector valued, it has a covariance matrix asso-
ciated with it, which is defined as
C
=
[(
−
[
])(
−
[
]) ]
T
.
(4.41)
EE E
θ
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