Problem Formulation - Study and Design of Differential Microphone Arrays - page 21

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 ◦

330 ◦

210 ◦ 330 ◦

210 ◦

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

60 ◦ 120 ◦ 60 ◦

120 ◦

150 ◦ 30 ◦ 150 ◦ 30 ◦

180 ◦

180 ◦ 0 ◦ 0 ◦

210 ◦ 330 ◦ 210 ◦ 330 ◦

240 ◦ 300 ◦ 240 ◦ 300 ◦

270 ◦ 270 ◦

(c) (d)

FIG. 2.3 Second-order directional patterns: (a) dipole, (b) cardioid, (c) hypercardioid, and
(d) supercardioid.

N

1
n ! x n + R N+1 ( x ) .

e x =

(2.23)

n=0

Substituting x = ( m − 1) ωτ 0 cos θ in (2.23) and neglecting the remainder,
we find that

N

1
n ! [( m − 1) ωτ 0 cos θ ] n .

e ( m − 1) ωτ 0 cos θ ≈

(2.24)

n=0

Using (2.24) in the general definition of the beampattern, we obtain