Study and Design of Differential Arrays with the MacLaurin’s Series Approximation - Study and Design of Differential Microphone Arrays - page 174

7.5 Adaptive Beamforming versus Differential Arrays

The linear system given in (7.3) can be generalized to

Ψ N+1 h ( ω )= a N+1 ( ω ) , (7.40)

where now

11 1 ··· 1
01 2 ··· M − 1
01 2 2 ··· ( M − 1) 2
. . . . . . .
012 N ··· ( M − 1) N

Ψ N+1 = (7.41)

is the constraint matrix of size ( N +1) ×M having the Vandermonde structure,
M is the number of microphones,

T

h ( ω )= H 1 ( ω ) H 2 ( ω ) ···H M ( ω ) (7.42)

is a filter of length M , and the vector a N+1 ( ω ) was already defined in (7.5).
In adaptive beamforming [2], we minimize the residual noise at the beam-
former output subject to the constraints summarized in (7.40). Mathemati-
cally, this is equivalent to

h(ω) h H ( ω ) Φ v ( ω ) h ( ω ) subject to Ψ N+1 h ( ω )= a N+1 ( ω ) . (7.43)

min

We easily deduce that the solution is the LCMV filter [2], [3]:

−1

−1
v

( ω ) Ψ N+1

−1
v

( ω ) Ψ N+1

h LCMV ( ω )= Φ Ψ N+1 Φ a N+1 ( ω ) . (7.44)

v ( ω ) Ψ N+1 in (7.44) to be full rank,
we must have N +1 ≤ M , which is the same condition to design a differential
array of order N .
For M = N + 1, we easily deduce from (7.44) that

−1

We observe that for the matrix Ψ N+1 Φ

−1

h LCMV ( ω )= Ψ

N+1 a N+1 ( ω ) ,

(7.45)

which corresponds exactly to the filter of an N th-order DMA or the solution
of (7.40).
For M>N + 1 and spatially white noise, (7.44) becomes

−1

h LCMV ( ω )= Ψ N+1 Ψ N+1 Ψ N+1

a N+1 ( ω ) , (7.46)

which corresponds to the minimum-norm solution of (7.40). This shows that
the LCMV filter is fundamentally related to the filter of an N th-order DMA.