Digital Signal Processing Reference
In-Depth Information
ˆ
β
ML
If f
(.)
is not reversible, then
is obtained as:
N
ˆ
ML
N
(
)
β
=
arg max
p
x
;
β
N
β
θ
(
)
()
(
)
p
x
;
β
=
ax
p
x
;
N
N
θβ
;
=
f
θ
Theorem 3.2 is a direct consequence of theorem 3.1; it stipulates that the ML
estimator achieves the Cramér-Rao bounds (∀
N
)
in the case where an efficient
estimator definitely exists. For example, this is the case when the signal follows a
linear model
xH b b
is a random Gaussian vector of known
covariance matrix. Moreover, theorem 3.3 shows that the ML estimator is
asymptotically optimum. For example, in the case of
()
=
θ
+
where
N
N
N
N
dN
=
N
this signifies that:
(
)
dist
(
)
ˆ
ML
N
−
1
()
N
θ
−
θ
.
0
,
F
θ
0
Finally, theorem 3.4 makes it possible to obtain the maximum likelihood
estimator of a function of θ.
This theorem can also be applied when it is difficult to
find or directly implement
ˆ
β
ML
N
ˆ
ML
θ To
sum up, the ML estimator possesses optimality properties for finite
N
as well as
asymptotically. For a wide range of problems, it can be implemented; in particular, it
is often systematically used in the case of deterministic signals buried in additive
Gaussian noise of known covariance matrix. In the latter case, the ML estimator
corresponds to a non-linear least squares estimator. In fact, let us suppose that the
signal is distributed as:
but it turns out to be simpler to obtain
.
(
)
()
x
N
s
θ
,
R
N
N
with known
R
N
. Then, the log-likelihood may be written as:
1
T
(
)
−
1
()
()
L
x
;
θ
=−
⎡
x
−
s
θ
⎤
R
⎡
x
−
s
θ
⎤
⎣
⎦
⎣
⎦
N
N
N
N
N
N
2
As a result, the maximum likelihood estimator of
θ
takes the following form:
T
ˆ
ML
N
−
1
()
()
θ
=
arg min
⎡
xs
−
θ
⎤
Rxs
⎡
−
θ
⎤
⎣
⎦
⎣
⎦
N
N
N
N
N
θ
which corresponds to the best approximation in the least squares sense of the data
x
N
by the model
s
N
(
θ
). In this case we end up with a relatively simple expression. On
the contrary,
when the covariance matrix is unknown and depends on
θ
, that is to
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