Study and Design of Second-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 84

90 ◦

0 dB

120 ◦ 60 ◦

−10 dB

−20 dB

150 ◦ 30 ◦

−30 dB

−40 dB

180 ◦ 0 ◦

210 ◦ 330 ◦

240 ◦ 300 ◦

270 ◦
FIG. 4.24 Directional pattern of the second-order cardioid with one distinct null.

2

1

2 + 1

B

C,2 ( θ )=

2 cos θ

. (4.35)

◦

This cardioid has only one null at 180 , while the one studied in the previous

section has two distinct nulls (at 90 ◦

◦

and 180 ). Figure 4.24 shows a plot of

B C,2 ( θ ) in dB.
We can use our approach to design such a DMA by choosing three con-
straints. The first two are a one at 0 ◦ and a null at 180 ◦ . From a theoretical
point of view, any value on the pattern can play the role of the third con-
straint but because of the approximations involved, this value must be chosen
with care in order that the derived filter corresponds to the original design.
Here we take the angle 3 60 ◦ (cos60 ◦ =1 / 2), so that B

C,2 (60 ◦ )=9 / 16.

Therefore, the linear system to solve is
d H ( ω, 1)

1

ω, 1
2

9
1 0

d H

h ( ω )= . (4.36)

d H ( ω,− 1)

3 Even with this value, we do not get exactly what we are looking for but the obtained
directional pattern is also very interesting. This shows, again, the flexibility of the approach.
In fact, taking the constraint above 90 ◦ , will lead exactly to the desired pattern. We leave
this study to the reader.