Minimum-Norm Solution for Robust Differential Arrays - Study and Design of Differential Microphone Arrays - page 113

which corresponds exactly to the filter of an N th-order DMA or the solution
of (6.1), which is a linear system of N + 1 equations and N + 1 unknowns.
For M>N + 1 and spatially white noise, (6.8) becomes

−1

h LCMV ( ω )= D H ( ω,Α ) D ( ω,Α ) D H ( ω,Α ) Β, (6.11)

which corresponds to the minimum-norm solution of (6.1). This shows that
the LCMV filter is fundamentally related to the filter of an N th-order DMA.
Even though we minimize the residual noise at the beamformer output,
there is no guaranty with the LCMV filter that the white noise gain should
be greater than 1; this is due to the multiple constraints that need to be
fulfilled at the same time. However, by virtue of the inclusion principle [6],
the solution for M>N + 1 is better than the solution for M = N + 1 when
it comes to white noise amplification. In fact, the greater is M , the better
is the solution. Therefore, contrary to what it may be said in the literature,
we can build a high-order DMA without amplifying much the white noise by
having more microphones as compared to the order of the DMA.
We can continue our comparisons to the minimum variance distortionless
response (MVDR) filter [7], [8]:

−1

v ( ω ) d ( ω, 1)
d H ( ω, 1) Φ

Φ

h MVDR ( ω )=

(6.12)

− v ( ω ) d ( ω, 1)

or, equivalently,

−1

y ( ω ) d ( ω, 1)
d H ( ω, 1) Φ

Φ

h MVDR ( ω )=

− y ( ω ) d ( ω, 1) .

(6.13)

This filter reduces more noise than the LCMV filter and the gain in SNR
is always greater than or equal to 1 (for all types of noise) [9] but Φ v ( ω )
or Φ y ( ω ) needs to be estimated accurately; otherwise, the estimation errors
may lead to some desired signal cancellation. Moreover, the pattern of the
MVDR filter does not correspond, in general, to any known directional pat-
tern contrary to the LCMV filter. The only exception is when Φ v ( ω ) in (6.12)
is replaced by Γ DN ( ω ). In this case, indeed, the MVDR filter corresponds to
the pattern of the hypercardioid (of order M − 1) but the white noise can
be highly amplified since when the number of microphones is increased so is
the order of the hypercardioid. This optimization procedure maximizes the
directivity factor at the expense of white noise amplification. While in the
LCMV filter, we can increase the number of microphones without increasing
much the order of the DMA; as a result, white noise amplification can be
controlled. We will get back to this point in the next section.
This study also suggests that adaptive beamforming with a linear array
should be greatly better in terms of noise reduction when the desired source
signal propagates from the endfire than any other directions. This way, we
also take advantage of the nature of DMAs.