Introduction - Study and Design of Differential Microphone Arrays - page 2

v M ( k ) v 1 ( k )

θ

M 2 1

δ

y M ( k ) y 2 ( k ) y 1 ( k )

FIG. 1.1 Illustration of a uniformly spaced linear additive microphone array for sound
capture in the farfield.

Additive arrays originally refer to the arrays that achieve signal enhance-
ment and noise suppression based on the synchronize-and-add principles; but
they have now evolved to include all the arrays with large inter-element spac-
ing (from a couple of centimeters to a couple of decimeters) and optimal gain
in broadside directions. In such arrays, each sensor converts the acoustic pres-
sure from its own viewpoint into an electrical signal. By processing all the
sensors' outputs, the signal of interest can be extracted while the unwanted
noise and interference can be suppressed. To illustrate how an additive array
works, let us consider a simple example with a uniformly spaced linear array
of M microphones as shown in Fig. 1.1. Assume that there is a single desired
source in the farfield such that its spherical wavefront appears planar at the
array. If we neglect the propagation attenuation, the signal received at the
discrete-time index k and at the m th microphone can be written as

y m ( k )= x m ( k )+ v m ( k ) (1.1)

= x ( k − t − τ m )+ v m ( k ) ,m =1 , 2 ,...,M,

where t is the propagation time from the source x ( k ) to microphone 1 (refer-
ence point), τ m is the relative time delay between the m th microphone and
the first one, and x m ( k )= x ( k − t − τ m ) and v m ( k ) are, respectively, the sig-
nal of interest and noise observed at the m th microphone. The noise signal,
v m ( k ), is assumed to be uncorrelated with x ( k ). With a uniform linear array
and a farfield source, the delay τ m can be expressed into the following form
according to the geometry shown in Fig. 1.1:

τ m = ( m − 1) δ cos θ
c

,m =1 , 2 ,...,M, (1.2)

where δ is the spacing between two neighboring microphones, c is the speed of
sound in the air, and θ is the signal incident angle. For simplicity of exposition,