Digital Signal Processing Reference
In-Depth Information
It is now interesting to compare the IF of
w
C
(
) to the general expression of the IF of
any
w
-MVDR functional
w
w
(
) derived in [48] and stated below.
Theorem 4 With the notations as in Theorem 3, the influence function of w-MVDR
functional w
w
(
)
at a CES distribution F
S
¼
CE
k
(
S
,
g
)
is given by
a
w
(
r
2
)
s
w
[
wa
H
I
]
S
1
=
2
uu
H
S
1
=
2
w
IF(
t
;
w
w
;
F
S
)
¼
where s
w
is the solution to
(2.27)
and
"
#
d
2
s
2
w
w
(
x
=
s
w
)
x
1
þc
w
1
k
(
k þ
1)
E w
0
d
s
w
a
w
(
x
)
¼
;
c
w
¼
(2
:
38)
and dis a positive real rva with the pdf
(2.12)
.
Theorem 4 shows that the IF of
w
w
(
) is continuous and bounded if, and only if,
w
(
x
)
x
is continuous and bounded. This follows by noting that when
ktk
, or equivalently
r ¼kS
1
=
2
tk
, grow to infinity,
u ¼ S
1
=
2
t=r
remains bounded. Hence, to validate
the qualitative robustness of
w
-MVDR beamformers we only need to validate that
w
(
x
)
x
is bounded. Theorem 4 also shows that IF(
a
;
w
w
,
F
S
)
¼
0, that is, if the contami-
nation point
t
equals the array response
a
, then it causes zero influence on the func-
tional. We wish to point out that if
w
w
(
) is the conventional MVDR functional
(i.e.,
w
;
1), then the IF expression of Theorem 4 indeed gives the IF expression of
Theorem 3. For example,
w
w
based on HUB(0.9) or MLT(1) functionals are robust,
that is, they have continuous and bounded IFs, since their
w
(
) functions are down-
weighting observations with large magnitude as shown in Figures 2.4 and 2.5.
We wish to point out that in beamforming literature, “robust” more commonly
refers to
robustness to steering errors
(imprecise knowledge of the array response
a
may be due to uncertainty in array element locations, steering directions and cali-
bration errors) and
robustness in the face of insufficient sample support
that may
lead to rank deficient SCM or inaccurate estimates of the array covariance matrix.
Thus, the lack of robustness is caused by misspecified or uncertain system matrices
or due to the fact that we do not have sufficient sample support to build up the rank
of the array covariance and pseudocovariance matrices, not because of uncertainty
in the probability models.
The diagonal loading of the SCM is one of the most popular techniques to over-
come the problems in the modeling system matrix or rank deficiency. Then we use
(
ˆ
CþgI
),
g
[ R
, in place of
ˆ
C
, which may not be full rank and hence not invertible.
For this type of robustness study, see for example, [12, 16, 23, 35] and references
therein. Here the term “robust” refers to statistical robustness to outliers [26], com-
monly measured by the concept of the IF. We wish to point out that robustness (as
measured by the IF) of the MVDR beamformer remains unaltered by diagonally load-
ing the covariance matrx
C
, that is, using
C
g
(
F
)
¼C
(
F
)
þgI
, where
g
is some constant
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