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T ,
T
B
BC 1 B
BC 2 B
; ...
matrix
such that the products
are as diagonal as possible
T
(subscript
indicates matrix transposition). Given an appropriate choice of the
diagonalization set {
B
is indeed an estimation of the
separating matrix in ( 8.2 ) and one obtains an estimate of the mixing matrix as
A ¼ B
C 1, C 2 ,
}, such matrix
þ . Matrices in {
} are chosen so as to hold in the off-diagonal entries
statistics describing some form of correlation among the sensor measurement
channels; then, the AJD will vanish those terms resulting in linear combination
vectors (the rows of
C 1, C 2 ,
B
) extracting uncorrelated components from the observed
mixture via ( 8.2 ). More particularly, the joint diagonalization is applied on matrices
that change according to the assumptions about the source. They are those changes,
when available, that provide enough information to solve the BSS problem. For-
mally, for AJDC, the identi
ability of sources discussed above, that is, matching
condition ( 8.3 ), is described by the fundamental AJD-based BSS theorem (Afsari
2008 ; see also A
ï
S 1 ,
S 2 ,
be the
K (unknown) covariance matrices of sources corresponding to the covariance
matrices included in the diagonalization set and s k ( ij ) their elements. The diagonal
elements of these matrices
ssa-El-Bey et al. 2008 ): Let matrices
s k ( ii ) hold the source variance. The off-diagonal elements
s ki ðÞ ; i 6 ¼ j
, are null as sources are assumed to be uncorrelated. Let
0
@
1
A
s 11 ðÞ ... s k 1 ðÞ
.
.
.
Þ T ¼
Y ¼ y 1 y M
ð
. .
ð 8 : 5 Þ
s 1 MM
s kMM
ð
Þ
ð
Þ
be the matrix formed by stacking one below the other row vectors
y 1 ,
y 2 ,
… y M
y m =( s 1( mm ) ,
, s K ( mm ) ) holds the
constructed as shown in Fig. 8.1 . Each vector
energy pro
M and
M the number of estimated sources. The fundamental theorem says that the m th
source can be separated as long as its energy pro
le along the diagonalization set for each source, with m :1
y m is not collinear 3
with any other vector in Y . Said differently, the wider the angle between y m and any
other vector in
le vector
, the greater the chance to separate the m th source. Even if two
vectors are collinear, the other sources can still be identi
Y
ed.
Table 8.1 reports useful information to de
ne an appropriate diagonalization set
so as to ensure identi
ability of sources.
Importantly, the two basic theoretical frameworks for working in a SOS
framework reported in Table 8.1 , the coloration and the non - stationary , can be
combined in any reasonable way: One may estimate covariance matrices in different
blocks (and/or conditions) for different frequency band-pass regions, effectively
increasing the uniqueness of the source energy pro
le. This is for instance the path
we have followed for solving the problem of separating sources generating error
potentials, as we will demonstrate here below. In fact, AJDC method can be applied
3
Two vectors are collinear if they are equal out of a scaling factor, that is, the energy pro le is
proportional.
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