Study and Design of First-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 38

3.3 First-Order Cardioid

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In the first-order cardioid, there is a one at the angle 0 and a null at the

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angle 180 (i.e., α 1,1 = − 1). Therefore, our linear system of two equations is

1 e ωτ 0
1 e

1
0

h ( ω )= . (3.19)

− ωτ 0

We easily find that the solution is

1
1 − e 2 ωτ 0

1
−e ωτ 0

h ( ω )= . (3.20)

Using (3.6) to approximate 1 − e 2ωτ 0 , we obtain

h ( ω ) ≈ 1
2 ωτ 0

1
−e ωτ 0

, (3.21)

from which we deduce the equivalent filter:

( ω )= 1
ω

1
−e ωτ 0

′

h ,

(3.22)

2 = 2τ 0 is a constant across frequencies. Taking the complex conjugate
of the components of h

C

since

′

( ω ), we get the gains that should be applied at the
two microphone outputs:

1 ( ω )= 1

′∗

H

ω ,

(3.23)

2 ( ω )= − 1

− ωτ 0 , ′∗

H

ω e

(3.24)

− ωτ 0 is simply a time-domain delay τ 0 at the second mi-
crophone output. Figure 3.6 illustrates the implementation of the first-order
cardioid.
The beampattern of the beamformer h

where the term e

′

( ω ) is

( ω ) ,θ ]= 1
ω

1 − e ωτ 0 (1+cos θ )

′

B [ h . (3.25)

Figure 3.7 displays the patterns from (3.25) for several frequencies and two
values of δ . Applying the approximation (3.6) to (3.25) leads to

( ω ) ,θ ] ≈ C

1

2 + 1

′

B [ h

2 cos θ

, (3.26)

which is the pattern of the first-order cardioid.