Digital Signal Processing Reference
In-Depth Information
Chapter 3
Study and Design of First-Order
Differential Arrays
This chapter is dedicated to the study and design of first-order differential mi-
crophone arrays. We explore the dipole, cardioid, subcardioid, hypercardioid,
and supercardioid.
3.1 Principle
In this work, the proposed approach to design DMAs is based on the obvious
observation that any ideal frequency-independent directional pattern has a
one at the angle
θ
=0
◦
and a number of nulls in some specific directions.
First-order DMAs are designed with two microphones. In this case, we have
exactly two constraints to fulfill. The first constraint is the distortionless
response (a one at the angle
θ
=0
◦
) and the second constraint is a null in
the interval 0
◦
<θ≤
180
◦
. Thus, these two constraints can be written as
d
H
(
ω,
cos0
◦
)
h
(
ω
)=
d
H
(
ω,
1)
h
(
ω
)=1
,
(3.1)
d
H
(
ω,α
1,1
)
h
(
ω
)=0
,
(3.2)
where
α
1,1
= cos
θ
1,1
is given by design (a null at the angle
θ
1,1
) with
−
1
≤
α
1,1
<
1. We can express (3.1)-(3.2) as
d
H
(
ω,
1)
d
H
(
ω,α
1,1
)
1
0
h
(
ω
)=
.
(3.3)
The matrix involved in (3.3) is recognized to be a 2
×
2 Vandermonde matrix.
Therefore, to design any first-order DMA, we only need to solve (3.3) and
the obtained solution is the best we can get from a mathematical point of
view.
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