Study and Design of First-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 44

where − 1 ≤ α 1,1 < 1 and 0 ≤ β 1,1 ≤ 1. We immediately find that the unique
solution to (3.34) is

1 − β 1,1 e ωτ 1
− (1 − β 1,1 ) e

1
1 − e ωτ 0 (1 − α 1,1 )

h ( ω )= , (3.35)

− ωτ 2

where τ 1 = τ 0 (1 − α 1,1 ) and τ 2 = τ 0 α 1,1 . Approximating 1 −e ωτ 0 (1 − α 1,1 )
with (3.6), we get

1 − β 1,1 e ωτ 1
− (1 − β 1,1 ) e

C
1 − α 1,1

· 1
ω

h ( ω ) ≈ , (3.36)

− ωτ 2

from which we deduce the equivalent filter 1 :

1 − β 1,1 e ωτ 1
− (1 − β 1,1 ) e

( ω )= 1
ω

′

h . (3.37)

− ωτ 2

′

Taking the complex conjugate of the components of h ( ω ), we get the gains

that should be applied at the two microphone outputs:

− ωτ 1

1 ( ω )= 1 − β 1,1 e

H ′∗

, (3.38)

ω

2 ( ω )= − (1 − β 1,1 ) e ωτ 2
ω

′∗

H , (3.39)

− ωτ 1 and e ωτ 2 are time-domain delays τ 1 and −τ 2 at the
first and second microphone outputs, respectively. Figure 3.11 illustrates the
implementation of the first-order differential array.
The beampattern corresponding to the beamformer h

where the term e

′

( ω ) is

1 − β 1,1 e ωτ 1 − (1 − β 1,1 ) e ω ( τ 0 cos θ − τ 2 ) ( ω ) ,θ ]= 1

ω

′

B [ h (3.40)

and applying the approximation (3.6) to the previous expression leads to

( ω ) ,θ ] ≈ 1 − α 1,1
C

1 − 1 − β 1,1
1 − α 1,1

+ 1 − β 1,1

′

B [ h

1 − α 1,1 cos θ

,

(3.41)

which is recognized to be the pattern of the first-order DMA. There are five
interesting cases.
•

Dipole: α 1,1 = β 1,1 = 0 (see Section 3.2).

• Cardioid: α 1,1 = − 1 ,β 1,1 = 0 (see Section 3.3).

• Subcardioid [2], [3]: α 1,1 = − 1 ,β 1,1 =0 . 4.

Hypercardioid: α 1,1 = − 2 ,β 1,1 = 0. •

1 Notice that this filter is noncausal for Α 1,1 > 0. In this case, we need to add a processing
delay equal to Τ 0 Α 1,1 = Τ 2 .