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. 2.5
Gain in SNR as a function of the direction (
Θ
N
) of the point noise source for the
first-order DMA: (a) dipole, (b) cardioid, (c) hypercardioid, and (d) supercardioid.
2.4 Vandermonde Matrix
Given the definition of the steering vector and combining steering vectors for
different angles in a matrix, we obtain the Vandermonde structure. Therefore,
it is extremely useful to exploit the structure of this matrix.
A Vandermonde matrix of size
M × M
has the form:
1
v
1
v
1
···v
M−1
1
···v
M−1
2
.
1
v
2
v
2
1
v
3
v
3
···v
M−1
3
V
M
=
(2.50)
.
.
.
.
.
.
.
···v
M−1
M
1
v
M
v
2
M
It can be shown that the determinant of
V
M
is
det (
V
M
)=
(
v
j
− v
i
)
.
(2.51)
j>i
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