Digital Signal Processing Reference
In-Depth Information
θ
2
1
·
τ
1
−τ
2
1 −
β
1
,
1
β
1,1
−
×
×
+
−
+
1
Ω
FIG. 3.11
Implementation of the first-order differential array.
Supercardioid:
α
1,1
=
1−
√
2
•
2
,β
1,1
= 0.
Figures 3.12, 3.13, and 3.14 display the patterns from (3.40) of the subcar-
dioid, hypercardioid, and supercardioid, respectively, for several frequencies
and two values of
δ
.
It can be verified that the white noise gain is
2−
√
(
ω
)] = 1
−
(1
− β
1,1
)
cos[
ω
(
τ
0
− τ
2
)]
β
1,1
+ (1
− β
1,1
)
2
− β
1,1
cos(
ωτ
1
)
− τ
2
)]
− β
1,1
cos[
ω
(
τ
0
− τ
1
′
G
WN,1
[
h
(3.42)
and for all the patterns where
β
1,1
= 0, (3.42) simplifies to
′
G
WN,1
[
h
(
ω
)] = 1
−
cos[
ω
(
τ
0
− τ
2
)]
=1
−
cos[
ωτ
0
(1
− α
1,1
)]
.
(3.43)
′
Figures 3.15 and 3.16 show plots of
G
(
ω
)] from (3.43), as a function of
frequency, for the hypercardioid and supercardioid, respectively, for different
values of
δ
.
For small values of
ω
(
τ
0
WN,1
[
h
− τ
2
), we can express (3.43) as
(
ω
)]
≈
1
2
[
ωτ
0
(1
− α
1,1
)]
2
.
′
G
WN,1
[
h
(3.44)
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