Minimum-Norm Solution for Robust Differential Arrays - Study and Design of Differential Microphone Arrays - page 122

6.3.2 SECOND-ORDER CARDIOID

From Chapter 4 on second-order DMAs, we find that the parameters of the
three fundamental constraints for the second-order cardioid are

T

Α = 1 − 10 , (6.23)

T .

Β = 100 (6.24)

In Figs 6.6, 6.7, and 6.8, we plot the patterns of the second-order cardioid
with 3, 5, and 8 microphones, respectively, for several frequencies. We observe
some minor differences when the number of microphones increases but all the
patterns correspond roughly to the second-order cardioid.
In Fig. 6.9, we give plots of the white noise gain of the second-order car-
dioid, as a function of frequency, for different values of M . As expected, the
white noise gain increases as M increases but the maximum is reached only
at high frequencies and for M ≥ 7. Comparing the 3 and 5 microphone cases
at 1 kHz, we observe that the gain is − 19 dB with 5 microphones and − 30 dB
with 3 microphones, which represents an improvement of 11 dB. For M = 8,
the white noise gain is only − 9 dB at 1 kHz.
Figure 6.10 shows plots of the directivity factor of the second-order car-
dioid, as a function of frequency, for different values of M . Comparing the
3 and 5 microphone cases, we see that the directivity factor is roughly the
same in both cases for frequencies up to 4 kHz; for higher frequencies, the
gain is a bit worst for M = 5.