Digital Signal Processing Reference
In-Depth Information
where the term 1
/ω
in (3.8) is simply a first-order low-pass filter. Figure 3.1
illustrates the implementation of the first-order dipole.
Now, the beampattern is
′
(
ω
)
,θ
]=
d
H
(
ω,
cos
θ
)
h
′
B
[
h
(
ω
)
=
1
ω
1
− e
ωτ
0
cos
θ
.
(3.9)
Figure 3.2 displays the patterns from (3.9) for several frequencies and two
values of
δ
. Applying the approximation (3.6) to (3.9) leads to
(
ω
)
,θ
]
≈
C
cos
θ.
′
B
[
h
(3.10)
Obviously, we recognize the pattern of the first-order dipole since 1
/C
is
simply a constant.
The white noise gain is
2
′H
(
ω
)
d
(
ω,
1)
h
′
G
WN,1
[
h
(
ω
)] =
′H
(
ω
)
h
′
h
(
ω
)
=
1
2
1
− e
ωτ
0
2
=1
−
cos(
ωτ
0
)
.
(3.11)
′
Figure 3.3 gives plots of
G
WN,1
[
h
(
ω
)] from (3.11), as a function of frequency,
for different values of
δ
.
The white noise is amplified if
′
G
WN,1
[
h
(
ω
)]
<
1
.
(3.12)
We observe from (3.11) that the sensor spacing must be chosen large, espe-
cially for low frequencies, if we don't want to amplify the white (or sensor)
noise. But a large value of
δ
is in contradiction with the DMA assumption,
which states that
δ
should be small. Therefore, there is always a tradeoff be-
tween white noise amplification (especially at low frequencies) and frequency-
independent directional pattern at high frequencies. Hence, the sensor spac-
ing should be selected according to this compromise. To better illustrate this
aspect, let us approximate (3.11) for small values of
ωτ
0
. We get
(
ω
)]
≈
(
ωτ
0
)
2
2
′
G
WN,1
[
h
.
(3.13)
√
We see that the gain is greater than 1 if
ωτ
0
>
2, which is somewhat in
contradiction with the condition given in (2.2).
The directivity factor (or the gain in SNR with diffuse noise) is
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