Digital Signal Processing Reference
In-Depth Information
5
5
4
4
3
3
2
2
1
1
0
0
0
0
0
.
5 1
.
0 1
.
5
2
.
0 2
.
5 3
.
0
3
.
5
4
.
0
0
.
5 1
.
0
1
.
5 2
.
0 2
.
5
3
.
0 3
.
5 4
.
0
F
(kHz)
F
(kHz)
(
a
)
(
b
)
5
5
4
4
3
3
2
2
1
1
0
0
0
0
0
.
5 1
.
0 1
.
5 2
.
0 2
.
5 3
.
0 3
.
5 4
.
0
0
.
5 1
.
0 1
.
5 2
.
0 2
.
5 3
.
0 3
.
5 4
.
0
F
(kHz)
F
(kHz)
(c)
(d)
FIG. 3.4
The directivity factor of the first-order dipole, 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.
which corresponds exactly, as expected, to the theoretical value of the direc-
tivity factor for the dipole with diffuse noise [1].
For a point noise source, the gain is
2
′H
(
ω
)
d
(
ω,
1)
h
′
G
NS,1
[
h
(
ω
)] =
′H
(
ω
)
d
(
ω,
cos
θ
N
)
|
2
|h
|
1
− e
ωτ
0
|
2
=
2
1
− e
ωτ
0
cos
θ
N
1
−
cos(
ωτ
0
)
1
−
cos(
ωτ
0
cos
θ
N
)
.
=
(3.17)
Therefore, for
θ
N
=0
◦
,
G
NS,1
[
h
′
(
ω
)] = 1
, ∀f
and for
θ
N
= 90
◦
,
G
NS,1
[
h
′
(
ω
)] =
′
∞, ∀f
. Figure 3.5 shows
G
NS,1
[
h
(
ω
)], as a function of
θ
N
, for several frequen-
cies and two values of
δ
. With the conventional approximation, we find that
1
cos
2
θ
N
,
′
G
NS,1
[
h
(
ω
)]
≈
(3.18)
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