Introduction - Study and Design of Differential Microphone Arrays - page 3

let us work in the frequency domain. Then, the signal model in (1.1) becomes

Y m ( ω )= X m ( ω )+ V m ( ω )

(1.3)

− ω ( t + τ m ) + V m ( ω ) ,m =1 , 2 ,...,M,

= X ( ω ) e

where ω =2 πf is the angular frequency, f denotes the temporal frequency,
=

√ − 1 represents the imaginary unit, and Y m ( ω ), X m ( ω ), V m ( ω ), and
X ( ω ) are the frequency-domain representations of y m ( k ), x m ( k ), v m ( k ), and
x ( k ), respectively.
Now, let us process the M signals Y m ( ω ) ,m =1 , 2 ,...,M , in order to
extract the desired signal X ( ω ) (up to a delay) and reduce the effect of V m ( ω ).
The most straightforward and simple way of doing this is through the use
of the so-called delay-and-sum (DS) beamformer. The basic principle of this
approach is that it compensates y m ( k ) with a delay τ m to align all the M
microphone signals, i.e., multiply Y m ( ω ) with e ωτ m , and then average the
results together. The DS beamformer output is then

M

Z ( ω )= 1
M

Y m ( ω ) e ωτ m

(1.4)

m=1

M

1
M

− ωt +

V m ( ω ) e ωτ m .

= X ( ω ) e

m=1

To check whether the beamformer output is less noisy than its input, let
us compare the input and output signal-to-noise ratios (SNRs). The input
SNR of the DS beamformer, according to the signal model given in (1.3), is
defined as the SNR at the first (reference) microphone, i.e.,

iSNR( ω )= φ X 1 ( ω )
φ V 1 ( ω )

(1.5)

= φ X ( ω )

φ V 1 ( ω ) ,

, φ V 1 ( ω )= E , and φ X ( ω )= |X 1 ( ω ) | 2 |V 1 ( ω ) | 2

where φ X 1 ( ω )= E

|X ( ω ) | 2 E , with E [ · ] denoting mathematical expectation. The output SNR

of the beamformer, from (1.4), is then

φ X ( ω )

oSNR( ω )= . (1.6)

1
M 2 E

2

m=1 V m ( ω ) e ωτ m

M

Now, we evaluate the following two cases.
•

The noise signals at the different microphones are uncorrelated.