Problem Formulation - Study and Design of Differential Microphone Arrays - page 17

where a N,n ,n =0 , 1 ,...,N , are real coecients. The different values of
these coecients determine the different directional patterns of the N th-order
DMA. In the direction of the desired signal, i.e., for θ =0

◦

, the beampattern

must be equal to 1, i.e., B N (0

◦

) = 1. Therefore, we have

N

a N,n =1 . (2.6)

n=0

As a result, we always choose the first coecient as

a N,0 =1 − N a N,n . (2.7)

n=1

It follows from (2.5) that an N th-order DMA has at most N (distinct) nulls.
All interesting patterns have at least one null in some direction. Since cos θ is
an even function, so is B N ( θ ). Therefore, on a polar plot, B N ( θ ) is symmetric
about the axis 0 ◦ − 180 ◦ and any DMA design can be restricted to this range.
Polar patterns are a very convenient way to describe the directional sensitivity
of the DMAs.
The directivity factor (see also Section 2.3) of an N th-order DMA, defined
as the ratio between the directivity pattern at the endfire direction θ =0

◦

and the averaged directivity pattern over the whole space, is 2 [4], [5], [6]

B N (0 ◦ )

G N =

π

1
π

B N ( θ ) dθ

0

π

= (2.8)

2

N

π

a N,n cos n θ

dθ

0

n=0

and what we call the directivity index is

D N = 10 log 10 G N . (2.9)

We find that the first-order, second-order, and third-order directivity factors
are

1
a 1,0 + 1

G

1 =

, (2.10)

2 a 1,1

1

G

2 =

, (2.11)

a 2,0 + 1

2 a 2,1 + 3 8 a 2,2 + a 2,0 a 2,2