Digital Signal Processing Reference
In-Depth Information
The solution to (4.36) is
1
1
−
ωτ
0
/
2
− e
ωτ
0
e
ωτ
0
/
2
h
(
ω
)=
e
−e
e
ωτ
0
/
2
− e
ωτ
0
−
ωτ
0
/
2
− e
3
ωτ
0
/
2
1
9
16 (1
− e
ωτ
0
)
−
ωτ
0
− e
ωτ
0
1
+
−e
.
(4.37)
−
ωτ
0
1
− e
Using the approximation
e
x
≈
1+
x
in the two factor terms of (4.37), we
obtain
1
−e
ωτ
0
−
16
−
ωτ
0
/
2
+
9
−
ωτ
0
h
(
ω
)
≈−
7
16
ω
2
τ
0
7
e
7
e
,
(4.38)
−
9
7
+
16
7
e
ωτ
0
/
2
from which we deduce the equivalent filter
4
:
1
−
ωτ
0
−
16
7
e
ωτ
0
/
2
+
9
7
e
ωτ
0
(
ω
)=
1
ω
2
−e
′∗
h
−
9
7
+
16
−
ωτ
0
/
2
7
e
1
1
=
1
ω
−
16
7
e
ωτ
0
/
2
+
9
7
e
ωτ
0
ω
0
0
1
−
ωτ
0
ω
−
e
.
(4.39)
−
16
7
e
ωτ
0
/
2
+
9
7
e
ωτ
0
Figure 4.25 shows how this cardioid can be implemented.
It can be verified that the beampattern of the beamformer
h
′
(
ω
) is
1
− e
ωτ
0
(cos
θ
+ 1)
(
ω
)
,θ
]=
1
ω
2
′
B
[
h
×
16
7
e
−
ωτ
0
/
2
−
9
−
ωτ
0
e
ωτ
0
cos
θ
1
−
7
e
.
(4.40)
Figure 4.26 displays the patterns from the previous equation for several fre-
quencies and two values of
δ
. For small values of
ωτ
0
, we can approximate
4
Notice that this filter is noncausal. Therefore, we need to add a processing delay equal
to
Τ
0
.
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