Minimum-Norm Solution for Robust Differential Arrays - Study and Design of Differential Microphone Arrays - page 128

6.3.3 THIRD-ORDER CARDIOID

It can be checked that the parameters of the four fundamental constraints to
design a third-order cardioid are summarized into the two vectors:

T

1 − 10 − √

Α =

2
2

, (6.25)

100 − √

T

Β =

2
8

+ 4

. (6.26)

Figures 6.11, 6.12, and 6.13 display the patterns of the third-order cardioid
with 4, 5, and 8 microphones, respectively, for several frequencies. All patterns
look similar.
In Fig. 6.14, we give plots of the white noise gain of the third-order car-
dioid, as a function of frequency, for different values of M . Comparing the 4
and 8 microphone cases at 1 kHz, we observe that the gain is − 50 dB with
4 microphones and around − 24 dB with 8 microphones, which represents an
improvement of about 26 dB.
In Fig. 6.15, we show plots of the directivity factor of the third-order
cardioid, as a function of frequency, for different values of M . Comparing the
4 and 8 microphone cases, we see that the gain in the second design is only
slightly worse in the range 3-6 kHz.