Study and Design of First-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 54

80 80

60 60

40 40

20 20

0 0

0 ◦ 60 ◦ 120 ◦ 180 ◦ 240 ◦ 300 ◦ 360 ◦

Θ N

0 ◦ 60 ◦ 120 ◦ 180 ◦ 240 ◦ 300 ◦ 360 ◦

Θ N

( a ) ( b )

80 80

60 60

40 40

20 20

0 0

0 ◦ 60 ◦ 120 ◦ 180 ◦ 240 ◦ 300 ◦ 360 ◦

Θ N

0 ◦ 60 ◦ 120 ◦ 180 ◦ 240 ◦ 300 ◦ 360 ◦

Θ N

( c ) ( d )

80 80

60 60

40 40

20 20

0 0

0 ◦ 60 ◦ 120 ◦ 180 ◦ 240 ◦ 300 ◦ 360 ◦

Θ N

0 ◦ 60 ◦ 120 ◦ 180 ◦ 240 ◦ 300 ◦ 360 ◦

Θ N

(e) (f)

FIG. 3.20 The gain of the first-order hypercardioid for a point noise source, as a function
of Θ N , for several frequencies and two values of Δ : (a) Ω =0 . 125 Π , Δ = 1 cm, (b) Ω =0 . 125 Π ,
Δ = 2 cm, (c) Ω =0 . 5 Π , Δ = 1 cm, (d) Ω =0 . 5 Π , Δ = 2 cm, (e) Ω = Π , Δ = 1 cm, and (f)
Ω = Π , Δ = 2 cm.

we can only form any directional patterns at the angles 0 ◦ and 180 ◦ . This
problem is due to the fact that the beampattern of the beamformer h

′

( ω )
[or the steering vector, d ( ω, cos θ )] is an even function of the variable θ . As a
result, the main lobe of a directional pattern at a direction different from 0 ◦
(or 180 ◦ ) will be affected because of the symmetry about the axis 0 ◦ − 180 ◦
on a polar plot.