Digital Signal Processing Reference
In-Depth Information
fact, overlapping for SCM and HUB(0.9). The estimated spectrums associated with
the different estimators are overlapping in the Gaussian case, so they provide essen-
tially the same DOA estimates. In the Cauchy noise case (setting B), however, the
conventional MVDR fails completely and can not resolve the two sources: the esti-
mated beampattern and spectrum are flat and the mainlobe and the peaks cannot be
well identified. Beampatterns associated with MLT(1) and HUB(0.9), however,
show a narrow mainlobe centered at the look direction and a deep null at DOA
of interference. Also spectrums for MLT(1) and HUB(0.9) show two sharp
peaks at the DOAs of the sources. Hence the performance loss is negligible by
employing MLT(1) or HUB(0.9) instead of the SCM in nominal Gaussian noise
conditions. However, significant gain in performance is obtained when the noise
is heavy-tailed Cauchy.
2.7.1 The Influence Function Study
First we derive the IF of the conventional MVDR functional. First note that the con-
ventional MVDR functional can be written in the form
w
C
(
F
)
¼C
(
F
)
1
g
(
F
)
where
a
g
(
F
)
W
a
H
C
(
F
)
1
a
is the normalized steering vector that satisfies
kC
(
F
)
1
=
2
g
(
F
)
k¼
1.
Applying the product rule of differentiation on the identity
C
(
F
1
,
t
)
C
1
(
F
1
,
t
)
¼ I
shows that
IF(
t
;
C
1
,
F
)
¼C
1
IF(
t
;
C
,
F
)
C
1
where
C
is the value of the covariance matrix at the reference distribution
F
.
Substituting the expression (2.17) for the IF of the covariance matrix
C
(
) into the
result above yields the following expression
IF(
t
;
C
1
,
F
)
¼C
1
tt
H
C
1
þC
1
(2
:
36)
for the IF of the inverse of the covariance matrix.
Now, using the product rule of differentiation, the IF of the conventional MVDR
functional
w
C
can be split into two parts
IF(
t
;
w
C
,
F
)
¼
@
@1
C
(
F
1
,
t
)
1
g
(
F
1
,
t
)
j
1¼
0
¼
IF(
t
;
C
1
,
F
)
gþC
1
IF(
t
;
g
,
F
)
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