Digital Signal Processing Reference
In-Depth Information
implemented no matter if (7.2) holds or not, while the filter
h
(
ω
) is considered
as the ideal filter for which (7.2) holds.
The beampattern of the beamformer
h
′
(
ω
) is
′
(
ω
)
,θ
]=
d
H
(
ω,
cos
θ
)
h
′
B
[
h
(
ω
)
=
1
ω
−
ωτ
0
a
1,0
/a
1,1
+
e
ωτ
0
cos
θ
−e
,
(7.10)
therefore,
2
ω
2
ωτ
0
a
1,1
(
ω
)
,θ
]
|
2
=
′
|B
[
h
1
−
cos
,
(7.11)
and taking
1
a
1,1
=1
− α
1,1
(7.12)
in (7.11), we obtain the exact same patterns as the ones obtained in Chapter 3
for the dipole, cardioid, hypercardioid, and supercardioid.
It is of interest to verify that, indeed, when (7.2) holds, we have
B
[
h
(
ω
)
,θ
]=
a
1,0
+
a
1,1
cos
θ.
(7.13)
′
The white noise gain of the beamformer
h
(
ω
) is
2
′H
(
ω
)
d
(
ω,
1)
h
′
G
WN,1
[
h
(
ω
)] =
h
′H
(
ω
)
h
′
(
ω
)
=
1
2
2
−
ωτ
0
a
1,0
/a
1,1
− e
ωτ
0
e
ωτ
0
a
1,1
=1
−
cos
.
(7.14)
Taking again (7.12) in (7.14), we obtain the exact same figures as the ones
obtained in Chapter 3 for the dipole, cardioid, hypercardioid, and supercar-
dioid.
For the beamformer
h
(
ω
), the white noise gain is
2
h
H
(
ω
)
d
(
ω,
1)
G
WN,1
[
h
(
ω
)] =
h
H
(
ω
)
h
(
ω
)
(
ωτ
0
)
2
2
a
1,1
+(
ωτ
0
)
2
a
1,0
=
.
(7.15)
We conclude that the white noise is amplified if
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