Study and Design of First-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 32

θ

2 1

δ

−

+

1
Ω

FIG. 3.1 Implementation of the first-order dipole.

3.2 First-Order Dipole

◦ ◦

The first-order dipole has a one at the angle 0 and a null at the angle 90

(i.e., α 1,1 = 0). Hence, the system of two linear equations (3.3) becomes

1 e ωτ 0
11

1
0

h ( ω )= , (3.4)

for which the solution is

1
1 − e ωτ 0

− 1

h ( ω )= . (3.5)

We recall that τ 0 = δ/c . Assuming that the sensor spacing is much smaller
than the acoustic wavelength, we can use the approximation:

e x ≈ 1+ x (3.6)

in (3.5) to get

h ( ω ) ≈ 1
ωτ 0

− 1

. (3.7)

The frequency-independent constant C = τ 0 has no importance; therefore,
h ( ω ) is simplified to the equivalent filter:

( ω )= 1
ω

− 1

′

h , (3.8)