Digital Signal Processing Reference
In-Depth Information
Θ
3
2
1
.
Δ
Δ
−τ
0
α
2,1
−τ
0
α
2,1
−
−
+
+
−τ
0
α
2,2
−
+
1
Ω
2
FIG. 4.6
Implementation of the second-order differential array.
1
−
1
−
cos(
ωτ
0
)
cos(
ωτ
0
)
(
ω
)=
1
ω
2
′
h
.
(4.25)
We recall that in the previous section, we approximated cos(
ωτ
0
) with
1; while this is possible for the dipole, it is not possible to approximate
cos
√
with 1 for the quadrupole, otherwise we get the dipole. Also,
it is of interest to notice that the dipole given in (4.25) should be better than
the one given in (4.8) in terms of the directional gain.
It is easy to see that the beampattern of the beamformer from (4.22) is
2
ωτ
0
/
2
1
− e
ωτ
0
(cos
θ − α
2,1
)
1
− e
ωτ
0
(cos
θ − α
2,2
)
(
ω
)
,θ
]=
1
ω
2
′
B
[
h
.
(4.26)
Figures 4.7, 4.8, 4.9, and 4.10 display the patterns from (4.26), for the car-
dioid, hypercardioid, supercardioid, and quadrupole, respectively, for several
frequencies and two values of
δ
. We can approximate (4.26) as the frequency-
independent second-order directional pattern:
(
ω
)
,θ
]
≈
C
2
(cos
θ − α
2,1
) (cos
θ − α
2,2
)
.
′
B
[
h
(4.27)
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