Problem Formulation - Study and Design of Differential Microphone Arrays - page 22

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. 2.4 Third-order directional patterns: (a) dipole, (b) cardioid, (c) hypercardioid, and
(d) supercardioid.

M

H m ( ω ) e ( m − 1) ωτ 0 cos θ

B [ h ( ω ) ,θ ]=

m=1

M N

1

n ! [( m − 1) ωτ 0 cos θ ] n

≈ H m ( ω )

m=1 n=0

N

( ωτ 0 ) n
n !

M

cos n θ ( m − 1) n H m ( ω )

≈

n=0 m=1

N

a N,n cos n θ = B

≈

N ( θ ) ,

(2.25)

n=0

where