Digital Signal Processing Reference
In-Depth Information
50
50
40
40
30
30
20
20
10
10
0
0
0
◦
60
◦
120
◦
180
◦
240
◦
300
◦
360
◦
0
◦
60
◦
120
◦
180
◦
240
◦
300
◦
360
◦
Θ
N
Θ
N
(a)
(b)
50
50
40
40
30
30
20
20
10
10
0
0
0
◦
60
◦
120
◦
180
◦
240
◦
300
◦
360
◦
0
◦
60
◦
120
◦
180
◦
240
◦
300
◦
360
◦
Θ
N
Θ
N
(c)
(d)
FIG. 2.6
Gain in SNR as a function of the direction (
Θ
N
) of the point noise source for the
second-order DMA: (a) dipole, (b) cardioid, (c) hypercardioid, and (d) supercardioid.
As a consequence, as long as the values of
v
m
are all distinct, the matrix
V
M
is nonsingular.
It will be important to have a closed-form expression of the inverse of
the Vandermonde matrix. For that, we will use the decomposition proposed
in [8]:
−1
V
M
=
U
M
L
M
,
(2.52)
where
U
M
and
L
M
are upper and lower triangular matrices, respectively.
The elements
l
ij
of
L
M
are given by the relations:
0
,
i<j
1
,
i
=
j
=1
l
ij
=
.
(2.53)
1
v
j
− v
p
,
otherwise
i
p=1,p
=j
It is proved in [8] that the elements
u
ij
of
U
M
are given by the definition:
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