Study and Design of Third-Order Differential Arrays with Three Distinct Nulls - Study and Design of Differential Microphone Arrays - page 109

which can be approximated as the frequency-independent N th-order direc-
tional pattern

N

( ω ) ,θ ] ≈ C N

′

B [ h (cos θ − α N,n ) . (5.18)

n=1

We can show that the white noise gain is

N

{ 1 − cos[ ωτ 0 (1 − α N,n )] }

′ n=1

G WN,N [ h ( ω )] = , (5.19)

1+ 1
2

cos[ ωτ 0 ( α N,j − α N,i )]

i j>i

j>i cos[ ωτ 0 ( α N,j

where − α N,i )] = 0 for N = 1. For N> 1, we can ap-

proximate (5.19) as

( ωτ 0 ) 2N N (1 − α N,n ) 2

n=1

′

G WN,N [ h

( ω )] ≈

.

(5.20)

3 × 2 N−1

We deduce that the white noise is amplified if

1
2N

3 × 2 N−1

ωτ 0 <

1/N .

(5.21)

N

(1 − α N,n )

n=1

We also find that the directivity factor and the gain for a point noise source
are, respectively,

N

1
ω 2N

2

1 − e ωτ 0 (1 − α N,n )

n=1

′

G

DN,N [ h

( ω )] = (5.22)

h ′H ( ω ) Γ DN ( ω ) h ′ ( ω )

and

N

{ 1 − cos[ ωτ 0 (1 − α N,n )] }

n=1

′

G

NS,N [ h

( ω )] = . (5.23)

N

{ 1 − cos[ ωτ 0 (cos θ N − α N,n )] }

n=1