Digital Signal Processing Reference
In-Depth Information
7.4. Link with a maximum likelihood estimator
By comparing the form of the equations, Capon [CAP 69] and Lacoss [LAC 71]
concluded that the minimum variance method was equivalent to a maximum
likelihood method. Since 1977, Lacoss recognized the “strange” meaning of this
terminology, but he wrote that it is “too late” to change it [LAC 77]. Eleven years
later, Kay examined the difference between these two estimators again [KAY 88].
Let us consider the simple case where the signal
x
(
k
)
is a complex exponential
function at the frequency
f
exp
embedded in an additive complex white, Gaussian,
centered noise
b
(
k
) of variance σ
2
.
Let C
c
be the complex amplitude of this
exponential, the signal is described in the following vector form:
XE B
[7.30]
=
C
+
cf
exp
where
and
B
are defined in equation [7.7].
XE
,
f
exp
If the frequency
f
exp
is supposed to be known, it is possible to calculate an
estimation C
c
of the complex amplitude C
c
by the maximum likelihood method.
Because the noise
b
(
k
)
is Gaussian, maximizing the probability density of
(
−
XE
means minimizing the quantity
(
)
)
H
(
)
−
1
C
XE
−
C
RXE
−
C
cf
cf
b
cf
exp
exp
exp
with
R
b
the noise correlation matrix. The minimization leads to the following result
[KAY 88]:
−
1
RE
bf
C
H
exp
=
HX
with
H
=
[7.31]
c
H
f
−
1
ERE
b
f
exp
exp
The bias and the variance of C
c
are written:
()
C
H
[7.32]
E
=
HE
C
=
C
c
c
f
c
exp
1
()
⎛
⎛
()
2
⎞
⎞
ˆ
ˆ
ˆ
H
var C
=
E
C
−
E
C
=
HRH
=
[7.33]
⎜
⎟
⎜
⎟
c
c
c
b
H
f
−
1
⎝
⎠
⎝
⎠
ERE
b
f
exp
exp
The maximum likelihood estimator can be interpreted as the output of a filter
whose input is the signal
x
(
k
)
and of impulse response
H
. Another way of
proceeding leads to the same expression of the estimator
C
c
. It is about minimizing
the filter output variance given by the first part of equation [7.33] by constraining
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