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g
i
P
G
i
g
i
G
i
r
i
Figure 6.4
Geometry of the Cramer-Rao bound in the separable statistical model with
multivariate Gaussian errors; the variance is large when mode
g
i
lies near the subspace
G
i
of
other modes.
of
. We shall find that it is the sensitivity of noise-free measurements to small variations
in parameters that determines the performance of an estimator.
The partial derivatives are
∂
y
/∂θ
i
=
h
i
, where
h
i
is the
i
th column of
H
and the
M
H
H
R
−
1
Fisher matrix is
J
F
=
nn
H
. The Cramer-Rao bound for frequentist-unbiased
estimators is thus
1
M
(
H
H
R
−
1
nn
H
)
−
1
≥
.
Q
(
)
(6.56)
R
−
1
/
2
nn
With the definition
G
=
H
,the(
i
,
i
)th element may be written as
g
i
g
i
1
M
1
g
i
g
i
1
M
1
g
i
g
i
1
sin
2
≥
P
G
i
)
g
i
=
ρ
i
,
Q
ii
(
)
(6.57)
g
i
(
I
−
where
P
G
i
is the projection onto the subspace
G
i
spanned by all but the
i
th mode in
G
,
ρ
i
is the angle that the mode vector
g
i
makes with this subspace, and
g
i
(
g
i
g
i
)
is the sine-squared of this angle. Thus, as illustrated in Fig.
6.4
, the lower bound on
the variance in estimating
(
I
−
P
G
i
)
g
i
/
θ
i
is a large multiple of (
M
g
i
g
i
)
−
1
when the
i
th mode can
be linearly approximated with the other modes in
G
i
. For closely spaced modes, only
a large number of independent samples or a large value of
g
i
g
i
- producing a large
output signal-to-noise ratio
M
g
i
g
i
- can produce a small lower bound. With low output
signal-to-noise ratio and closely spaced modes, any estimator of
θ
i
will be poor, meaning
that the resolution in amplitude of the
i
th mode will be poor. This result generalizes to
mean-value vectors more general than
H
, on replacing
h
i
with
∂
/∂θ
i
.
Example 6.2.
Let the noise covariance matrix in the proper multivariate Gaussian model
be
R
nn
=
σ
2
I
and the matrix
H
e
j
φ
k
e
j(
n
−
1)
φ
k
]
T
.We
=
[
A
1
,
A
2
] with
k
=
[1
,
,...,
call
k
a complex exponential mode with mode angle
φ
k
and
A
a complex mode
amplitude. The Cramer-Rao bound is
1
M
1
1
Q
ii
(
)
≥
φ
1
−
φ
2
)
,
(6.58)
n
|
A
|
2
/σ
2
1
−
l
n
(
where
sin
2
(
n
1
n
2
φ/
2)
l
n
(
0
≤
φ
)
=
2)
≤
1
(6.59)
sin
2
(
φ/
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