Many situations are either not well suited to linear estimation
procedures, or the parameter is not well described as a random
variable. For example, signal delay is observed nonlinearly and
usually no a priori density can be assigned.
In such cases, maximum likelihood estimators are more frequently
used. Because of the Cramér-Rao
bound, fundamental limits on parameter estimation
performance can be derived for any signal
parameter estimation problem where the parameter is not random.
Assume that the data are expressed as a signal observed in the
presence of additive Gaussian noise.
∀l,l∈0…L−1:rl=slθ+nl
l
l
0
…
L
1
r
l
s
l
θ
n
l
(1)
The vector of observations
rr is formed from the data in the
obvious way. Evaluating the logarithm of the observation
vector's joint density,
lnpr|θr=-1/2lndet2π
K
n
−1/2r−sθT
K
n
-1r−sθ
p
r
θ
r
-12
2
K
n
12
r
s
θ
K
n
r
s
θ
where
sθ
s
θ
is the signal vector having
P
P unknown parameters, and
K
n
K
n
is the covariance matrix of the noise. The partial
derivative of this likelihood function with respect to the
i
th
i
th
parameter
θ
i
θ
i
, for real-valued signals,
∂∂
θ
i
lnpr|θr=r−sθT
K
n
-1∂∂
θ
i
sθ
θ
i
p
r
θ
r
r
s
θ
K
n
θ
i
s
θ
and, for complex-valued ones,
∂∂
θ
i
lnpr|θr=ℜr−sθH
K
n
-1∂∂
θ
i
sθ
θ
i
p
r
θ
r
r
s
θ
K
n
θ
i
s
θ
If the maximum of the likelihood function can be found by
setting its gradient to
0
0, the maximum likelihood estimate of the parameter
vector is the solution of the set of equations
∀i,i∈1…P:r−sθT
K
n
-1∂∂
θ
i
sθ|θ,θ=θ̂ML=0
i
i
1
…
P
θ
θ
ML
r
s
θ
K
n
θ
i
s
θ
0
(2)
The Cramér-Rao bound depends on the evaluation of the Fisher
information matrix
F
F. The elements of this matrix are found to be
∀i,j,i∧j∈1…P:Fij=∂∂
θ
i
sθT
K
n
-1∂∂
θ
j
sθ
i
j
i
j
1
…
P
F
i
j
θ
i
s
θ
K
n
θ
j
s
θ
(3)
Further computation of the Cramér-Rao bound's components
is problem dependent if more than one parameter is involved, and
the off-diagonal terms of
F
F are nonzero. If only one parameter is unknown, the
Cramér-Rao bound is given by
Eε2≥b2θ+1+ddθbθ2∂∂θsθT
K
n
-1∂∂θsθ
ε
2
b
θ
2
1
θ
b
θ
2
θ
s
θ
K
n
θ
s
θ
(4)
When the signal depends on the parameter nonlinearly (which
constitute the interesting cases), the maximum likelihood
estimate is usually biased. Thus, the numerator of the
expression for the bound cannot be ignored. One interesting
special case occurs when the noise is white. The
Cramér-Rao bound becomes
Eε2≥b2θ+
σ
n
21+ddθbθ2∑l=0L−1∂∂θslθ2
ε
2
b
θ
2
σ
n
2
1
θ
b
θ
2
l
0
L
1
θ
s
l
θ
2
The derivative of the signal with respect to the parameter can
be interpreted as the sensitivity of the signal to the
parameter. The mean-squared estimation error depends on the
"integrated" squared sensitivity: The greater this
sensitivity, the smaller the bound.
For an efficient estimate of a signal parameter to exist, the
estimate must satisfy the condition we derived earlier.
∇θsθT
K
n
-1r−sθ≟I+∇θbT-1∇θsθT
K
n
-1∇θsθ
θr
̂−θ−b
≟
θ
s
θ
K
n
r
s
θ
I
θ
b
θ
s
θ
K
n
θ
s
θ
θ
r
θ
b
Because of the complexity of this requirement, we
quite rightly question the existence of any efficient estimator,
especially when the signal depends nonlinearly on the parameter
(see this
problem).
Let the unknown parameter be the signal's amplitude; the
signal is expressed as
θsl
θ
s
l
and is observed in an array's output in the presence
of additive noise. The maximum likelihood estimate of the
amplitude is the solution of the equation
r−θ̂MLsT
K
n
-1s=0
r
θ
ML
s
K
n
s
0
The form of this equation suggests that the maximum
likelihood estimate is efficient. The amplitude estimate is
given by
θ̂ML=rT
K
n
-1ssT
K
n
-1s
θ
ML
r
K
n
s
s
K
n
s
The form of this estimator is precisely that of the
matched filter derived in the colored-noise situation (see
equation). The
expected value of the estimate equals the actual amplitude.
Thus the bias is zero and the Cramér-Rao bound is given
by
Eε2≥sT
K
n
-1s-1
ε
2
s
K
n
s
The condition for an efficient estimate becomes
sT
K
n
-1r−θs≟sT
K
n
-1sθ̂ML−θ
≟
s
K
n
r
θ
s
s
K
n
s
θ
ML
θ
whose veracity we can easily verify.
In the special case where the noise is white, the estimator
has the form
θ̂ML=rTs
θ
ML
r
s
, and the Cramér-Rao bound equals
σ
n
2
σ
n
2
(the nominal signal is assumed to have unit energy).
The maximum likelihood estimate of the amplitude has
fixed error characteristics that do not
depend on the actual signal amplitude. A signal-to-noise
ratio for the estimate, defined to be
θ2Eε2
θ
2
ε
2
, equals the signal-to-noise ratio of the observed
signal.
When the amplitude is well described as a random variable, its
linear minimum mean-squared error estimator has the form
θ̂LIN=
σ
θ
2rT
K
n
-1s1+
σ
θ
2sT
K
n
-1s
θ
LIN
σ
θ
2
r
K
n
s
1
σ
θ
2
s
K
n
s
which we found in the white-noise case becomes a
weighted version of the maximum likelihood estimate (see example).
θ̂LIN=
σ
θ
2
σ
θ
2+
σ
n
2rTs
θ
LIN
σ
θ
2
σ
θ
2
σ
n
2
r
s
Seemingly, these two estimators are being used to
solve the same problem: Estimating the amplitude of a signal
whose waveform is known. They make very different
assumptions, however, about the nature of the unknown
parameter; in one it is a random variable (and thus it has a
variance), whereas in the other it is not (and variance makes
no sense). Despite this fundamental difference, the
computations for each estimator are equivalent. It is
reassuring that different approaches to solving similar
problems yield similar procedures.
