Introduction - Study and Design of Differential Microphone Arrays - page 4

If all the noise signals, V m ( ω ) ,m =1 , 2 ,...,M , are uncorrelated with
each other and have the same variance, it is easy to check that the gain in
SNR is

G ( ω )= oSNR( ω )

iSNR( ω ) = M.

(1.7)

In this scenario, we see that a simple DS beamformer can improve the SNR
by a factor of M or 10 log 10 M in dB and the gain in SNR is independent
of frequency.

• The noise signals are from a point source with an incident angle of θ N .

In this case, we have

− ω ( m − 1) δ cos θ N /c .

V m ( ω )= V 1 ( ω ) e (1.8)

Substituting V m ( ω ) into (1.6) and with some mathematical manipulations,
one can have

2

M sin [ ωδ (cos θ − cos θ N ) / (2 c )]
sin[ Mωδ (cos θ − cos θ N ) / (2 c )]

oSNR( ω ) = iSNR( ω ) . (1.9)

Therefore, the gain in SNR is

2

M sin [ ωδ (cos θ − cos θ N ) / (2 c )]
sin[ Mωδ (cos θ − cos θ N ) / (2 c )]

G ( ω )= . (1.10)

It is seen that G ( ω ) is a function of the number of sensors, the microphone
spacing, the angular separation between the signal of interest and the noise
signal to be suppressed, and frequency. Figure 1.2 plots the gain in SNR
with a uniform linear array of ten microphones.

Another way to study the performance of an additive array is through the
directivity pattern (also called beampattern). For the signals given in (1.3), if
we neglect the noise terms and compensate the m th microphone signal with
a delay equal to ( m − 1) δ cos θ S /c , where θ S is the angle of the desired source
signal, the DS beamformer output is

M

Z ( ω )= 1
M

− ωt e − ω ( m − 1) δ (cos θ − cos θ S ) /c .

X ( ω ) e (1.11)

m=1

The beampattern, which is defined as the magnitude of the transfer function
between the beamformer output and the input signal, is then

Z ( ω )
X ( ω )

sin [ Mωδ (cos θ − cos θ S ) / (2 c )]
M sin[ ωδ (cos θ − cos θ S ) / (2 c )]

B ( ω,θ )= = . (1.12)