Digital Signal Processing Reference
In-Depth Information
Chapter 4
Study and Design of Second-Order
Differential Arrays
In this chapter, we study and design the most important second-order dif-
ferential microphone arrays. We are concerned with the dipole, two types of
cardioid, quadrupole, hypercardioid, and supercardioid.
4.1 Principle
Any second-order DMA can be realized with three microphones. Therefore,
we assume that we have exactly three microphones. As a result, we have three
constraints to fulfill with the first one being, as usual, a one at the angle
θ
=0
◦
. We deduce that the general linear system of equations to design any
second-order differential array is
d
H
(
ω,
1)
d
H
(
ω,α
2,1
)
d
H
(
ω,α
2,2
)
1
β
2,1
β
2,2
h
(
ω
)=
,
(4.1)
where
−
1
≤ α
2,1
<
1,
−
1
≤ α
2,2
<
1,
α
2,1
=
α
2,2
,0
≤ β
2,1
≤
1, and
0
≤ β
2,2
≤
1.
Let us denote by
d
H
(
ω,
1)
d
H
(
ω,α
2,1
)
d
H
(
ω,α
2,2
)
V
3
=
1
v
1
v
1
1
v
2
v
2
1
v
3
v
3
=
(4.2)
the 3
×
3 Vandermonde matrix that appears in (4.1), where
v
1
=
e
ωτ
0
,
v
2
=
e
ωτ
0
α
2,1
, and
v
3
=
e
ωτ
0
α
2,2
. From the decomposition
V
−1
3
=
U
3
L
3
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