Digital Signal Processing Reference
In-Depth Information
10
5
0
10
5
0
−5
−10
−15
−20
−25
−30
−5
−10
−15
−20
−25
−30
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)
(
a
)
(
b
)
10
5
0
10
5
0
−5
−10
−15
−20
−25
−30
−5
−10
−15
−20
−25
−30
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. 3.17
The directivity factor of the first-order hypercardioid, 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.
(
ω
)] is maximized with
α
1,1
=
−
3
.
For our hypercardioid (
α
1,1
=
−
1
/
2) and the supercardioid, the gains are, re-
spectively, 27
/
7 and 3
/
′
It can also be easily verified that
G
DN,1
[
h
15
−
10
√
. Figure 3.19 shows a plot of
G
′
2
DN,1
[
h
(
ω
)]
from (3.52), as a function of
α
1,1
.
For
β
1,1
= 0, the gain for a point noise source is
1
−
cos[
ωτ
0
(1
− α
1,1
)]
1
−
cos[
ωτ
0
(cos
θ
N
′
G
NS,1
[
h
(
ω
)] =
− α
1,1
)]
,
(3.55)
′
where
G
NS,1
[
h
(
ω
)] =
∞, ∀f
for cos
θ
N
=
α
1,1
. Figures 3.20 and 3.21 show
′
plots of
G
(
ω
)], as a function of
θ
N
, for the hypercardioid and super-
cardioid, respectively, for several frequencies and two values of
δ
. For small
values of
ωτ
0
, (3.55) becomes
NS,1
[
h
1
′
G
NS,1
[
h
(
ω
)]
≈
2
,
(3.56)
α
1,1
1
− α
1,1
1
1
− α
1,1
+
cos
θ
N
which corresponds to the theoretical gain of the first-order DMA.
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