Study and Design of Second-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 59

Θ

3 2 1

.

Δ

Δ

− −

+ +

− +

1
Ω 2

FIG. 4.1 Implementation of the second-order dipole.

1
− 1 − cos( ωτ 0 )
cos( ωτ 0 )

h ( ω ) ≈− 1
ω 2 τ 0

− 2
1

≈− 1
ω 2 τ 0

.

(4.7)

Since C 2 = − τ 0 is a constant across frequencies, it is more convenient to
work with the equivalent filter:

− 2
1

( ω )= 1
ω 2

′

h , (4.8)

where the term 1 /ω 2 in (4.8) is simply a second-order low-pass filter. We can
express (4.8) as

− 1
0

0
− 1 ( ω )= 1

ω

1

− 1
ω

′

h , (4.9)

ω

showing that the second-order dipole can be implemented as a cascade of
first-order dipoles (three in total). Figure 4.1 illustrates the implementation
of the second-order dipole.
The beampattern of the beamformer h

′

( ω ) is