Digital Signal Processing Reference
In-Depth Information
where
a
i
denotes the
i
th component of the array response
a
. An informative picture
on the effect of contamination
t
1
¼ u
1
þ j
v
1
on
w
w
is obtained by the surface plot
of the norm of the empirical influence function
k
EIF(
t
;
w
w
,
Z
n
k
) with respect to
v
1
and
v
1
. The EIFs in Figure 2.9 are averages over 100 realizations. Sample lengths
are
n ¼
50, 500,
1
, where the surface plots under
n ¼1
correspond to the asymp-
totic value
k
IF(
t
,
w
w
,
F
4
)
k
. As expected, we observe that when the sample size
grows (from
n ¼
50 to
n ¼
500), the calculated EIF surfaces more accurately
resemble the corresponding theoretical IF surface. However, at the small sample
size (
n ¼
50), the relative influence of an additional observation on the estimator
is a bit larger than that the IF would indicate. The surface plots neatly demonstrate
the nonrobustness of the conventional MVDR beamformer for both the finite and
large sample cases: outlying points with large values of
u
1
and
/
or
v
1
have bounded
influence in the cases of HUB(0.9) or MLT(1) but large and unbounded influence
when the conventional SCM is employed.
Efficiency Study
Using the IF of
w
w
(
) (cf. Theorem 4) and equations (2.19) and
(2.20) as the definitions for the asymptotic covariance matrix and pseudo-covariance
matrix of the estimator, the next theorem was proved in [48].
Theorem 5 With the notations as in Theorem 4, the asymptotic covariance matrix of
the estimated w-MVDR weight w
w
when sampling from F
S
¼
CE
k
(
S
,
g
)
is
ASC(
w
w
;
F
S
)
¼ l
w
(
Gww
H
)
where
E
[
w
2
(
d
=
s
w
)(
d
=
s
w
)
2
]
k
(
k þ
1)(1
þc
w
)
2
l
w
¼
:
The asymptotic pseudo-covariance matrix of w
w
vanishes, that is,
ASP(
w
w
;
F
S
)
¼
0
.
Note that the ASC of
w
w
depends on the selected
M
-estimator and on the functional
form of the CES distribution
F
S
only
via the real-valued positive multiplicative con-
stant
l
w
. (Observe that the matrix term
Gww
H
does not depend on the choice of
w
and on
F
S
only via
S
.) Hence comparisons of this single scalar index is needed only. It
is a surprising result that ASP vanishes, which means that
w
w
has asymptotic circular
CN distribution.
Note also that ASC(
w
w
;
F
S
) is singular and of rank
k
1 (since the nullspace of
Gww
H
has dimension 1 due to MVDR constraint
w
H
a ¼
1, so (
Gww
H
)
a ¼
0).
Thus the asymptotic CN distribution of
w
w
is singular. This is an expected result
since singular distributions commonly arise in constrained parameter estimation
problems, where the constraint imposes a certain degree of deterministicity to the
estimator.
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