Digital Signal Processing Reference
In-Depth Information
′
(
ω
)
,θ
]=
d
H
(
ω,
cos
θ
)
h
′
B
[
h
(
ω
)
1
ω
2
2
1
− e
ωτ
0
cos
θ
=
.
(4.10)
′
Figure 4.2 displays the patterns of
h
(
ω
) from the above equation for several
frequencies and two values of
δ
. For small values of
ωτ
0
, we can approximate
(4.10) as
(
ω
)
,θ
]
≈
C
2
cos
2
θ,
′
B
[
h
(4.11)
where we recognize the pattern of the second-order dipole.
The white noise gain is
2
h
′H
(
ω
)
d
(
ω,
1)
′
G
WN,2
[
h
(
ω
)] =
′H
(
ω
)
h
′
h
(
ω
)
=
1
6
1
− e
ωτ
0
4
=
2
3
[1
−
cos(
ωτ
0
)]
2
.
(4.12)
In Fig. 4.3, we plot
G
WN,2
[
h
(
ω
)] from (4.12), as a function of frequency, for
different values of
δ
. For small values of
ωτ
0
, we get
′
(
ω
)]
≈
(
ωτ
0
)
4
6
′
G
WN,2
[
h
,
(4.13)
from which we deduce that the white noise is amplified if
ωτ
0
<
6
1/4
≈
1
.
57
.
(4.14)
Comparing this result with the one obtained for the first-order dipole (see
Chapter 3), we observe that the situation for the second-order dipole is worse.
The directivity factor is
2
h
′H
(
ω
)
d
(
ω,
1)
′
G
DN,2
[
h
(
ω
)] =
h
′H
(
ω
)
Γ
DN
(
ω
)
h
′
(
ω
)
|
1
− e
ωτ
0
|
4
2 [3
−
4sinc(
ωτ
0
) + sinc(2
ωτ
0
)]
=
2 [1
−
cos(
ωτ
0
)]
2
3
−
4sinc(
ωτ
0
) + sinc (
ωτ
0
) cos(
ωτ
0
)
.
=
(4.15)
′
In Fig. 4.4, we plot
G
(
ω
)] from (4.15), as a function of frequency, for
different values of
δ
. Using Taylor's series, we can show that
DN,2
[
h
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