Minimum-Norm Solution for Robust Differential Arrays - Study and Design of Differential Microphone Arrays - page 114

6.2 Minimum-Norm Filter

As it is clearly explained in the previous section, the minimum-norm solution
given in (6.11) is of great importance here since it has the potential to solve
the problem of white noise amplification encountered in all known differential
arrays. Let us slightly change the notation of this filter and denote it by

−1

h ( ω,Α, Β )= D H ( ω,Α ) D ( ω,Α ) D H ( ω,Α )

Β, (6.14)

where, again, the vectors Α and Β of length N + 1 determine the pattern
and the order of the DMA. Basically, the length of these vectors determine
(roughly) the order of the DMA while their values determine the pattern.
Meanwhile, the length, M , of the minimum-norm filter, h ( ω,Α, Β ),canbe
much larger than N + 1, which will help make the DMA robust against
white noise amplification. In our context, a more rigorous way to derive the
minimum-norm filter is by maximizing the white noise gain subject to the
N th-order DMA fundamental constraints. This is equivalent to minimizing
h H ( ω,Α, Β ) h ( ω,Α, Β ) subject to (6.1), from which (6.14) results. In other
words, the LCMV filter in the presence of white noise is the solution we
are looking for. Therefore, for a large number of microphones, the white
noise gain should approach M . If the value of M is much larger than N +
1, the order of the DMA may not be equal to N anymore but the N th-
order DMA fundamental constraints will always be fulfilled. As a result, the
resulting shape of the directional pattern may slightly be different than the
one obtained with M = N + 1. This approach is the best we can do to solve
the conflicting requirement of a high-order DMA that does not amplify the
white noise.
We suggest to use (6.14) for the design of any order DMA. Of course,
its implementation is going to be very different from the classical one where
only delays and a low-pass filter are needed. Here, instead, we should rely
on a subband or frequency-domain implementation, which can still be very
eŽcient from a complexity point of view.
It is easy to see that the beampattern of the minimum-norm filter is

B [ h ( ω,Α, Β ) ,θ ]= d H ( ω, cos θ ) h ( ω,Α, Β )
= d H ( ω, cos θ ) D H ( ω,Α )

−1

D ( ω,Α ) D H ( ω,Α ) Β.

(6.15)

We also deduce that the white noise gain, the directivity factor, and the
gain for a point noise source are, respectively,

1
Β T [ D ( ω,Α ) D H ( ω,Α )]

G

WN [ h ( ω,Α, Β )] =

, (6.16)

−1 Β