Minimum-Norm Solution for Robust Differential Arrays - Study and Design of Differential Microphone Arrays - page 134

6.3.4 THIRD-ORDER DMA

In this subsection, we propose to design with the minimum-norm filter, the
third-order DMA with three distinct nulls proposed in Chapter 5 as Case 1.
In this scenario, we have

T

10 − 1
2

Α = , (6.27)

− 1

T .

Β = 1000 (6.28)

Figures 6.16, 6.17, and 6.18 display the patterns of the third-order DMA
with three distinct nulls and 4, 5, and 8 microphones, respectively, for several
frequencies. All patterns look very similar except for the case M = 8 at high
frequencies.
In Fig. 6.19, we give plots of the white noise gain of the third-order DMA
with three distinct nulls, 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
around − 48 dB with 4 microphones and around − 23 dB with 8 microphones,
which represents an improvement of about 25 dB.
In Fig. 6.20, we show plots of the directivity factor of the third-order DMA
with three distinct nulls, 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 4-7 kHz.
In this chapter, we have shown the effectiveness of the minimum-norm
filter in the design of robust DMAs. Fundamentally, we have exploited the
fact that we have many more microphones than the order of the DMA. We
can further improve the robustness of the DMAs at low frequencies by using
the minimum-norm filter with a non-uniform linear array but we leave this
study to the reader.