Digital Signal Processing Reference
In-Depth Information
1
G
3
=
.
(2.12)
a
3,0
+
1
2
a
3,1
+
3
8
a
3,2
+
a
3,0
a
3,2
+
5
16
a
3,3
+
3
4
a
3,1
a
3,3
The hypercardioid is the pattern obtained from the maximization of the
directivity factor
3
.
The front-to-back ratio is defined as the ratio of the power of the output
of the array to signals propagating from the front-half plane to the output
power for signals arriving from the rear-half plane [7]. This ratio, for the
cylindrically isotropic noise field, is mathematically defined as [4], [7]
π/2
B
N
(
θ
)
dθ
0
F
N
=
.
(2.13)
π
B
N
(
θ
)
dθ
π/2
The supercardioid is the pattern obtained from the maximization of the front-
to-back ratio
4
[7].
First-order directivity patterns have the form:
B
1
(
θ
) = (1
− a
1,1
)+
a
1,1
cos
θ
(2.14)
and the most important ones are as follows.
•
Dipole:
a
1,1
= 1, null at cos
θ
= 0, and
D
1
= 3 dB.
Cardioid:
a
1,1
=
2
, null at cos
θ
=
−
1, and
D
•
1
=4
.
3 dB.
Hypercardioid:
a
1,1
=
3
, null at cos
θ
=
−
1
/
2, and
D
•
1
=4
.
8 dB.
√
√
√
•
Supercardioid:
a
1,1
=2
−
2, null at cos
θ
= (1
−
2)
/
(2
−
2), and
D
1
=4
.
6 dB.
Figure 2.2 shows these different polar patterns. What is exactly shown are
the values of the magnitude squared beampattern in dB, i.e., 10 log
10
B
1
(
θ
).
Second-order beampatterns are described by the equation:
− a
2,2
)+
a
2,1
cos
θ
+
a
2,2
cos
2
θ.
B
2
(
θ
) = (1
− a
2,1
(2.15)
The second-order dipole has a null at cos
θ
= 0 and a one (maximum)
at cos
θ
=
−
1. Replacing these values in (2.15), we find that
a
2,1
=0and
a
2,2
= 1. By analogy with the first-order and second-order dipoles, we define
the
N
th-order dipole as
B
D,N
(
θ
) = cos
N
θ,
(2.16)
3
Another type of hypercardioid can be obtained by maximizing the directivity factor in
the presence of a spherically isotropic noise field. There is not much difference, however,
between the two patterns.
4
Another type of supercardioid can be obtained by maximizing the front-to-back ratio in
the presence of a spherically isotropic noise field. There is not much difference, however,
between the two patterns.
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