Study and Design of Third-Order Differential Arrays with Three Distinct Nulls - Study and Design of Differential Microphone Arrays - page 94

C 3
ω 3 (1 − α 3,1 ) (1 − α 3,2 )(1 − α 3,3 )

h ( ω ) ≈ ×

1

− ωτ 0 α 3,1 − e − ωτ 0 α 3,2 − e − ωτ 0 α 3,3

−e

,

− ωτ 0 ( α 3,1 + α 3,2 ) + e − ωτ 0 ( α 3,2 + α 3,3 ) + e − ωτ 0 ( α 3,1 + α 3,3 )

e

− ωτ 0 ( α 3,1 + α 3,2 + α 3,3 )

−e

(5.5)

where the term 1 /ω 3 is a third-order low-pass filter. We deduce that the
equivalent filter is 1

( ω )= 1
ω 3

′∗

h ×

1
−e ωτ 0 α 3,1 − e ωτ 0 α 3,2 − e ωτ 0 α 3,3
e ωτ 0 ( α 3,1 + α 3,2 ) + e ωτ 0 ( α 3,2 + α 3,3 ) + e ωτ 0 ( α 3,1 + α 3,3 )
−e ωτ 0 ( α 3,1 + α 3,2 + α 3,3 )

.

(5.6)

Figure 5.1 illustrates the implementation of the third-order differential array.
It can be verified that the beampattern of the beamformer h

′

( ω ) is

3

( ω ) ,θ ]= 1
ω 3

1 − e ωτ 0 (cos θ − α 3,n )

′

B [ h , (5.7)

n=1

which can be approximated as the frequency-independent second-order di-
rectional pattern:

3

( ω ) ,θ ] ≈ C 3

′

B [ h (cos θ − α 3,n ) . (5.8)

n=1

Figures 5.2, 5.3, and 5.4 display the patterns from (5.7), of Case 1, Case 2,
and Case 3, respectively, for several frequencies and two values of δ .
The white noise gain is

1 Notice that this filter is noncausal for Α 3,1 > 0 or Α 3,2 > 0 or Α 3,3 > 0. In this situation,
a processing delay has to be added.