Digital Signal Processing Reference
In-Depth Information
Tabl e 5 . 2
BSS algorithms based on joint diagonalization (continued)
algorithm
source model
condition matrices
optimization
algorithm
SONS [52]
non-stationary
s
(
t
)
with diagonal (auto-
)covariances
(auto-)covariance
matrices of windowed
signals
orthogonal
JD
after
PCA
ACDC [292],
LSDIAG
[297]
independent or auto-
decorrelated
s
(
t
)
covariance ma-
trices and cumu-
lant/autocovariance
matrices
non-
orthogonal
JD
block-
Gaussian
likelihood
[203]
block-Gaussian
non-
(auto-)covariance
matrices of windowed
signals
non-
orthogonal
JD
stationary
s
(
t
)
TFS [27]
s
(
t
)f omC 's
time-frequency distri-
butions [58]
spatial time-
frequency distribution
matrices
orthogonal
JD
after
PCA
FRT-
based
non-stationary
s
(
t
)
with diagonal block-
spectra
autocovariance
of
(non-
)orthogonal
JD
BSS
FRT-transformed
windowed signal
[129]
ACMA [273]
s
(
t
) is of constant
modulus (CM)
independent
vectors
generalized
Schur
ker
P
of
QZ-
in
model-
matrix
P
decomp.
stBSS [254]
spatiotemporal
sources
s
:=
s
(
r, t
)
any of the above con-
ditions for both
x
and
x
non-
orthogonal
JD
group
BSS
group-dependent
sources
s
(
t
)
any of the above con-
ditions
block or-
thogonal JD
after PCA
[249]
model is then defined by requiring the sources to fulfill
C
i
(
s
)=0
for all
i
=1
,...,K
. In table 5.1, we review some commonly used
source conditions for an
m
-dimensional centered random vector
x
and a
multivariate random process
x
(
t
).
Searching for sources
s
:=
Wx
fulfilling the source model requires
finding matrices
W
such that
C
i
(
Wx
) is diagonal for all
i
. Depending
on the algorithm, whitening by PCA is performed as preprocessing
to allow for a reduced search on the orthogonal group
W
O
(
n
).
This is equivalent to setting all source second-order statistics to
I
,and
then searching only for rotations. In the case of
K
=1,thesearch
can be performed by eigenvalue decomposition of
C
1
(
x
) of the source
condition of the whitened mixtures
x
; this is equivalent to solving the
generalized eigenvalue decomposition (GEVD)
problem for the matrix
pencil (
E
(
xx
)
,
C
1
(
x
)). Usually, using more than one condition matrix
∈
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