Problem Formulation - Study and Design of Differential Microphone Arrays - page 19

90 ◦ 90 ◦

0 dB
−10 dB
−20 dB
−30 dB
−40 dB

0 dB
−10 dB
−20 dB
−30 dB
−40 dB

120 ◦ 60 ◦

120 ◦ 60 ◦

150 ◦ 30 ◦ 150 ◦ 30 ◦

180 ◦ 0 ◦

180 ◦ 0 ◦

210 ◦ 330 ◦ 210 ◦ 330 ◦

240 ◦ 300 ◦

240 ◦ 300 ◦

270 ◦ 270 ◦

(a) (b)

90 ◦ 90 ◦

0 dB
−10 dB
−20 dB
−30 dB
−40 dB

0 dB
−10 dB
−20 dB
−30 dB
−40 dB

120 ◦ 60 ◦ 120 ◦ 60 ◦

150 ◦ 30 ◦

150 ◦ 30 ◦

180 ◦ 0 ◦ 180 ◦ 0 ◦

210 ◦

330 ◦ 210 ◦ 330 ◦

240 ◦ 300 ◦ 240 ◦ 300 ◦

270 ◦ 270 ◦

(c) (d)

FIG. 2.2 First-order directional patterns: (a) dipole, (b) cardioid, (c) hypercardioid, and
(d) supercardioid.

implying that a N,N = 1 and a N,N−1 = a N,N−2 = ··· = a N,0 = 0. The
N th-order dipole has only one (distinct) null (in the range 0 ◦ − 180 ◦ ) at
θ = 90 ◦ . The directivity indices of the second-order and third-order dipoles
are, respectively, D

3 =5 . 1 dB.
The most well-known second-order cardioid has two nulls; one at cos θ =
− 1 and the other one at cos θ = 0. From these values, we easily deduce from
(2.15) that a 2,1 = a 2,2 = 2 . By analogy with the first-order and second-order
cardioids, we define the N th-order cardioid as

2 =4 . 3 dB and D

1

2 + 1

cos N−1 θ,

B C,N ( θ )=

2 cos θ

(2.17)

implying that a N,N = a N,N−1 = 2 and a N,N−2 = a N,N−3 = ··· = a N,0 = 0.
This N th-order cardioid has only two distinct nulls (in the range 0 ◦ − 180 ◦ ):