Study and Design of Third-Order Differential Arrays with Three Distinct Nulls - Study and Design of Differential Microphone Arrays - page 108

5.3 Generalization

In this section, we generalize the ideas of the third-order DMA to the N th
order. We seek to generate a directional pattern of order N , which has N
distinct nulls, with a uniform linear array of N +1= M microphones. As a
result, the linear system of N + 1 equations to generate this general pattern
is

d H ( ω, 1)
d H ( ω,α N,1 )
d H ( ω,α N,2 )
.
d H ( ω,α N,N )

1
0
0
.
0

h ( ω )= ,

(5.15)

where − 1 ≤ α N,1 ,α N,2 ,...,α N,N < 1 and α N,1 = α N,2 = ··· = α N,N .
It can be shown that the equivalent filter deduced from (5.15) is

( ω )= 1
ω N

T ,

′∗

1 h ′∗

2 ( ω ) h ′∗ 3 ( ω ) ···h ′∗

h

N+1 ( ω )

(5.16)

where

N

e ωτ 0 α N,i 1 ,

′∗

h

2 ( ω )= −

i 1 =1

e ωτ 0 ( α N,i 1 + α N,i 2 ) ,

′∗

h

3 ( ω )=

i 1 =i 2

e ωτ 0 ( α N,i 1 + α N,i 2 + α N,i 3 ) , h ′∗

4 ( ω )= −

i 1 =i 2 =i 3

.

e ωτ 0 ( α N,i 1 + ··· + α N,i N ) ,

n+1 ( ω )=( − 1) n

′∗

h

i 1 =··· =i N

.

N+1 ( ω )=( − 1) N e ωτ 0 ( α N,i 1 + ··· + α N,i N ) .

′∗

h

This beamformer can be implemented as a cascade of N(N+1)

2 first-order
DMAs [1]. Thus, the first stage has N first-order DMAs, the second stage
has N − 1 first-order DMAs, etc., and the last stage has one first-order DMA.
The beampattern of the beamformer h

′

( ω ) is

N

( ω ) ,θ ]= 1
ω N

1 − e ωτ 0 (cos θ − α N,n )

′

B [ h , (5.17)

n=1