Minimum-Norm Solution for Robust Differential Arrays - Study and Design of Differential Microphone Arrays - page 115

1
h H ( ω,Α, Β ) Γ DN ( ω ) h ( ω,Α, Β ) ,

G DN [ h ( ω,Α, Β )] = (6.17)

and

1
|B [ h ( ω,Α, Β ) ,θ N ] | 2 ,

G

NS [ h ( ω,Α, Β )] =

(6.18)

where θ N is the angle of the point noise source (see Chapter 2).
Ideally, we would like to have

G

DN [ h ( ω,Α, Β )] = G N ,

(6.19)

where G

N is the directivity factor of the frequency-independent N th-order
DMA (see Chapter 2) and

G

WN [ h ( ω,Α, Β )] ≥ 1 .

(6.20)

6.3 Design Examples

In this section, we show how to design several directional patterns of different
orders with the minimum-norm filter and a uniform linear array of up to eight
microphones. We only study four examples. Evidently, many more patterns
can be designed with the proposed approach. In the previous chapters, we
considered a bandwidth of 4 kHz; here, we use a bandwidth of 8 kHz in
order to have a better insight into this new method. In all examples, the
inter-element spacing is δ = 1 cm.

6.3.1 FIRST-ORDER CARDIOID

From Chapter 3, we know that the parameters to design the first-order car-
dioid are summarized into the two vectors of length 2:

T ,

Α = 1 − 1 (6.21)

T .

Β = 10 (6.22)

Figures 6.1, 6.2, and 6.3 display the patterns of the first-order cardioid with
2, 5, and 8 microphones, respectively, for several frequencies. The patterns for
the 2 and 5 microphones cases are similar except for 5 kHz, where the pattern
with 5 microphones is more directional than the one with 2 microphones.
With 8 microphones, we clearly see that for 4 and 5 kHz, we get the directional
patterns of the second-order cardioid, which is expected and interesting. In