Problem Formulation - Study and Design of Differential Microphone Arrays - page 15

Θ

1

M 2

Δ

Y M (Ω) Y 2 (Ω) Y 1 (Ω)

H ∗ M (Ω) H 2 (Ω) H 1 (Ω)

Σ

FIG. 2.1 A uniform linear microphone array with processing.

the sensor spacing, δ , is much smaller than the acoustic wavelength, λ , i.e.,
δ ≪ λ , implying that

ωδ
c

= ωτ 0 ≪ 2 π. (2.2)

The condition (2.2) easily holds for small values of δ and at low frequencies
but not at high frequencies. With this condition, spatial aliasing, which has
the negative effect of creating grating lobes (i.e., copies of the main lobe,
which usually points toward the desired signal), is also avoided [2].
We consider fixed directional beamformers 1 , like in DMAs, where the main
lobe is at the angle θ =0 ◦ (endfire direction) and the desired signal prop-
agates at the same angle. This position is optimal as will become clearer
later. Electronic steering (in the sense that the main lobe can be oriented to
any possible direction without affecting the shape of the beampattern) with
a uniform linear DMA is not really feasible but we will study some simple
possibilities.
As pointed out in [3], there is a fundamental difference between differential
arrays and filter-and-sum beamformers. In the latter category, the filters are
optimized in such a way that the microphone signals are aligned in order
to steer the main lobe in the direction of the desired signal, whereas in the
former category the gains are optimized to steer a number of nulls in some
specific directions.
The focus of this work is on the design, with small apertures, of beam-
formers whose beampatterns are very close to the ones obtained with “ideal”

1 The terms beamformer, beamforming, and beampattern may not be adequate in the
context of DMAs but we will still use them for convenience.