Problem Formulation - Study and Design of Differential Microphone Arrays - page 26

[ Γ v ( ω )] ij =[ Γ DN ( ω )] ij = sin[ ω ( j − i ) τ 0 ]

ω ( j − i ) τ 0
= sinc [ ω ( j − i ) τ 0 ] .

(2.44)

In this scenario, the gain in SNR, G DN [ h ( ω )], is called the directivity factor
and the directivity index is simply defined as [2], [4]

D [ h ( ω )] = 10 log 10 G DN [ h ( ω )] . (2.45)

With diffuse noise, the filter h ( ω ) is often found by maximizing the direc-
tivity factor. As a result, the optimal filter is given by (2.38).

• The noise comes from a point source at the angle θ N . In this case, the

pseudo-coherence matrix is

Γ v ( ω )= d ( ω, cos θ N ) d H ( ω, cos θ N ) ,

(2.46)

where

T

− ωτ 0 cos θ N ···e − ( M − 1) ωτ 0 cos θ N

d ( ω, cos θ N )= (2.47)

1 e

is the steering vector of the noise source. We observe from (2.46) that the
pseudo-coherence matrix is singular. In fact, this is the only possibility
where the gain in SNR, G NS [ h ( ω )], is not upper bounded and can go to
infinity. We deduce that this gain is

2

h H ( ω ) d ( ω, cos0 ◦ )

G

NS [ h ( ω )] =

|h H ( ω ) d ( ω, cos θ N ) | 2

1
|h H ( ω ) d ( ω, cos θ N ) | 2 .

= (2.48)

When the noise and desired signals come from the same direction, i.e.,
when θ N =0

◦

, then there is no possible gain, i.e., G

NS [ h ( ω )] = 1 , ∀h ( ω ).

We also deduce the gain of an N th-order DMA:

1
|B N ( θ N ) | 2 .

G

NS,N ( θ N )=

(2.49)

Figures 2.5, 2.6, and 2.7 depict this gain, as a function of the direction
of the noise, for the different first-order, second-order, and third-order
patterns (dipole, cardioid, hypercardioid, and supercardioid).