Digital Signal Processing Reference
In-Depth Information
9
a
2,1
8
a
2,2
−
a
2,0
a
2,2
−
ωτ
0
3
a
2,1
/
2
a
2,2
+
(
ωτ
0
)
2
e
.
(7.26)
h
(
ω
)=
−
a
2,2
(
ωτ
0
)
2
−
ωτ
0
a
2,1
/a
2,2
−
a
2,1
a
2,2
(
ωτ
0
)
2
−
2
e
−
ωτ
0
a
2,1
/
2
a
2,2
+
a
2,1
8
a
2,2
(
ωτ
0
)
2
e
2
=
−
a
2,2
τ
0
Since
C
is constant across frequencies, the equivalent filter is
.
(7.27)
9
a
2,1
8
a
2,2
−
a
2,0
a
2,2
−
ωτ
0
3
a
2,1
/
2
a
2,2
+
(
ωτ
0
)
2
e
(
ω
)=
1
ω
2
−
ωτ
0
a
2,1
/a
2,2
−
a
2,1
a
2,2
′
(
ωτ
0
)
2
h
−
2
e
−
ωτ
0
a
2,1
/
2
a
2,2
+
a
2,1
8
a
2,2
(
ωτ
0
)
2
e
As we did it in the previous section, we consider
h
(
ω
) as the ideal filter
for which (7.2) holds and
h
(
ω
) the practical beamformer that should be
implemented no matter if (7.2) holds or not.
It can be verified that
′
B
[
h
(
ω
)
,θ
]=
d
H
(
ω,
cos
θ
)
h
(
ω
)
3
H
n
(
ω
)
e
ω
(
n −
1)
τ
0
cos
θ
=
n=1
1+
ωτ
0
cos
θ −
(
ωτ
0
)
2
cos
2
θ
2
=
H
1
(
ω
)+
H
2
(
ω
)
1+2
ωτ
0
cos
θ −
2(
ωτ
0
)
2
cos
2
θ
+
H
3
(
ω
)
=
a
2,0
+
a
2,1
cos
θ
+
a
2,2
cos
2
θ,
(7.28)
while
3
n
(
ω
)
e
ω
(
n −
1)
τ
0
cos
θ
.
′
′
B
[
h
(
ω
)
,θ
]=
H
(7.29)
n=1
Figures 7.1, 7.2, 7.3, and 7.4 display the patterns from (7.29) for the dipole,
cardioid, hypercardioid, and supercardioid, respectively, for several frequen-
cies and two values of
δ
.
The white noise gain of the beamformer
h
′
(
ω
) is
2
n
(
ω
)
e
ω
(
n −
1)
τ
0
3
n=1
H
′
′
G
WN,2
[
h
(
ω
)] =
.
(7.30)
3
n
(
ω
)
|
2
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