Digital Signal Processing Reference
In-Depth Information
10
10
0
0
−
10
−
10
−
20
−
20
−
30
−
30
−
40
−
40
−
50
−
50
−
60
−
60
−
70
−
70
−
80
−
80
0
.
5
1
.
0
1
.
5
2
.
0
2
.
5 3
.
0 3
.
5
4
.
0
0
0
.
5
1
.
0 1
.
5 2
.
0 2
.
5 3
.
0
3
.
5 4
.
0
0
F
(kHz)
F
(kHz)
(
a
)
(
b
)
10
10
0
0
−
10
−
10
−
20
−
20
−
30
−
30
−
40
−
40
−
50
−
50
−
60
−
60
−
70
−
70
−
80
−
80
0
0
.
5 1
.
0 1
.
5 2
.
0 2
.
5 3
.
0 3
.
5 4
.
0
0
0
.
5 1
.
0 1
.
5 2
.
0 2
.
5 3
.
0 3
.
5 4
.
0
F
(kHz)
F
(kHz)
(c)
(d)
FIG. 4.11
The white noise gain of the second-order cardioid, as a function of frequency,
for different values of
Δ
: (a)
Δ
= 1 cm, (b)
Δ
= 2 cm, (c)
Δ
= 3 cm, and (d)
Δ
= 5 cm.
In particular, it can be verified from (4.27) that for the quadrupole,
B
2
[
h
(
ω
)
,θ
] has three identical maximums at the angles 0
◦
, 90
◦
, and 180
◦
.
The white noise gain is
′
2
2
1
− e
ωτ
0
(1
− α
2,1
)
1
− e
ωτ
0
(1
− α
2,2
)
′
G
WN,2
[
h
(
ω
)] =
(4.28)
4+2cos[
ωτ
0
(
α
2,1
− α
2,2
)]
{
1
−
cos[
ωτ
0
(1
− α
2,1
)]
}{
1
−
cos[
ωτ
0
(1
− α
2,2
)]
}
1+
2
cos[
ωτ
0
(
α
2,1
=
.
− α
2,2
)]
′
In Figs. 4.11, 4.12, 4.13, and 4.14, we plot
G
(
ω
)] from (4.28), for
the cardioid, hypercardioid, supercardioid, and quadrupole, respectively, as a
function of frequency, for different values of
δ
. The white noise gain in (4.28)
can be approximated as
WN,2
[
h
(
ω
)]
≈
(
ωτ
0
)
4
(1
− α
2,1
)
2
(1
− α
2,2
)
2
6
′
G
WN,2
[
h
.
(4.29)
We deduce that the white noise is amplified if
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