Study and Design of Third-Order Differential Arrays with Three Distinct Nulls - Study and Design of Differential Microphone Arrays - page 95

Θ

4 3 2 1

. .

Δ Δ Δ

−τ 0 α 3,1 −τ 0 α 3,1 −τ 0 α 3,1

− −

−

+

+ +

−τ 0 α 3,2 −τ 0 α 3,2

− −

+ +

−τ 0 α 3,3

−

+

1
Ω 3

FIG. 5.1 Implementation of the third-order differential array.

3

2

1 − e ωτ 0 (1 − α 3,n )

n=1

′

G

WN,3 [ h

( ω )] = (5.9)

8+4 cos[ ωτ 0 ( α 3,j − α 3,i )]

i j>i

3

{ 1 − cos[ ωτ 0 (1 − α 3,n )] }

n=1

= .

1+ 1
2

cos[ ωτ 0 ( α 3,j − α 3,i )]

i j>i

In Figs 5.5, 5.6, and 5.7, we plot the white noise gain from the previous
expression, of Case 1, Case 2, and Case 3, respectively, as a function of
frequency, for different values of δ . We can approximate (5.9) as

3

( ωτ 0 ) 6 (1 − α 3,n ) 2

n=1

′

G

WN,3 [ h

( ω )] ≈ . (5.10)

12