Problem Formulation - Study and Design of Differential Microphone Arrays - page 14

Chapter 2
Problem Formulation

In this chapter, we explain some important aspects of beamforming and differ-
ential arrays. The problem of a DMA design is formulated while we progress
in defining some useful concepts. We start with the definition of the steering
vector for a plane wave with the conventional anechoic farfield model. We
give the general definition of the beampattern as well as its expression for
directional arrays. We then derive the gain in signal-to-noise ratio (SNR),
which can be very useful in the evaluation of DMAs under different types of
noise. Finally, we discuss the Vandermonde matrix, which always appears,
explicitly or implicitly, in the design of DMAs.

2.1 Signal Model

We consider a source signal (plane wave) that propagates in an acoustic
environment (anechoic farfield model) at the speed of sound, i.e., c = 340 m/s,
and impinges on a uniform linear sensor array consisting of M omnidirectional
microphones, where the distance between two successive sensors is equal to δ
(see Fig. 2.1). The direction of the source signal to the array is parameterized
by the angle θ . In this scenario, the corresponding steering vector (of length
M ) is

T

− ωδ cos θ/c ···e − ( M − 1) ωδ cos θ/c

d ( ω, cos θ )=

1 e

T

1 ··· M−1

− ωτ 0 cos θ − ωτ 0 cos θ

= , (2.1)

1 e e

√ − 1 is the imaginary
unit, ω =2 πf is the angular frequency, f> 0 is the temporal frequency,
and τ 0 = δ/c is the delay between two successive sensors at the angle θ =0 ◦ .
The acoustic wavelength is λ = c/f . In DMAs [1], it is always assumed that

where the superscript T is the transpose operator, =