Digital Signal Processing Reference
In-Depth Information
Chapter 5
Study and Design of Third-Order
Differential Arrays with Three
Distinct Nulls
In this chapter, we study an important family of third-order differential mi-
crophone arrays. In this family, which includes the hypercardioid and su-
percardioid, the directional patterns have exactly three distinct nulls. This
feature simplifies the design of third-order DMAs. We will also see how this
approach can be easily generalized to the
N
th order.
5.1 Principle
Any third-order directional pattern with three distinct nulls can be generated
with a uniform linear array of four microphones [1], [2]. It is clear that the
linear system of four equations tailored for the derivation of a beamformer
with three distinct nulls is
d
H
(
ω,
1)
1
d
H
(
ω,α
3,1
)
d
H
(
ω,α
3,2
)
d
H
(
ω,α
3,3
)
0
0
0
h
(
ω
)=
,
(5.1)
where
−
1
≤ α
3,1
<
1,
−
1
≤ α
3,2
<
1,
−
1
≤ α
3,3
<
1, and
α
3,1
=
α
3,2
=
α
3,3
.
We denote by
d
H
(
ω,
1)
d
H
(
ω,α
3,1
)
d
H
(
ω,α
3,2
)
d
H
(
ω,α
3,3
)
V
4
=
1
v
1
v
1
v
1
1
v
2
v
2
v
2
1
v
3
v
3
v
3
1
v
4
v
4
v
4
=
(5.2)
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