Problem Formulation - Study and Design of Differential Microphone Arrays - page 16

∗

DMAs. For that, a complex weight, H

m ( ω ) ,m =1 , 2 ,...,M , is applied
at the output of each microphone, where the superscript

∗

denotes complex
conjugation. The weighted outputs are then summed together to form the
beamformer output as shown in Fig. 2.1. Putting all the gains together in a
vector of length M , we get

T .

h ( ω )= H 1 ( ω ) H 2 ( ω ) ···H M ( ω ) (2.3)

The objective then is to design such a filter for any directivity pattern of
any order. The approach taken here is based on the fundamental observation
that for all beampatterns of interest, some constraints must be fulfilled at
all frequencies given that the number of microphones is equal to M . In other
words, we select M fundamental constraints from a well-defined beampattern
of a DMA. For example, in the first-order dipole with two microphones, the
two fundamental constraints are a one at the angle 0 ◦ and a null at the angle
90 ◦ . Since we have two microphones and two constraints, we have a simple
linear system of two equations to solve. As a result, the obtained solution is
optimal from a mathematical point of view and the derived dipole is the best
we can get.
In the next two sections, we discuss some fundamental measures. We are
only interested in narrowband measures. The broadband measures can be
easily deduced from their respective narrowband counterparts.

2.2 Beampattern

Each beamformer has a pattern of directional sensitivity, i.e., it has different
sensitivities from sounds arriving from different directions. The beampattern
or directivity pattern describes the sensitivity of the beamformer to a plane
wave (source signal) impinging on the array from the direction θ . Mathemat-
ically, it is defined as

B [ h ( ω ) ,θ ]= d H ( ω, cos θ ) h ( ω )

(2.4)

M

H m ( ω ) e ( m − 1) ωτ 0 cos θ ,

=

m=1

where the superscript H is the transpose-conjugate operator.
The frequency-independent beampattern of an N th-order DMA is well
known. It is defined as [4]

N

a N,n cos n θ,

B

N ( θ )=

(2.5)

n=0