Digital Signal Processing Reference
In-Depth Information
parameter space, thus leading to computational savings and faster convergence. One
of the early kurtosis maximization algorithms for instantaneous BSS in real-valued
mixtures is based on an ingenious parametrization of the separating matrix allow-
ing dimensionality reduction at the deflation step [
26
], and can be summarized as
follows.
The method relies on a preliminary prewhitening step leading to linearly trans-
formed observations
z
(n)
N
with identity covariance matrix. Under the source
independence and unit-variance assumption, one can easily see that the whitened
observations are linked to the sources through an unknown orthogonal transforma-
tion
Q
∈ R
N
×
N
, resulting in the observation model
∈ R
z
=
Qs
.
(9.15)
Source separation is then achieved from the whitened observations through a par-
ticular deflation approach. This approach relies on the decomposition of matrix
Q
in terms of Givens planar rotations
Q
i,j
(θ)
, defined as an identity matrix except for
entries
(i,i)
,
(i,j)
,
(j,i)
, and
(j,j)
,1
≤
i<j
≤
N
, which are given by
cos
θ
sin
θ
−
.
sin
θ
cos
θ
More precisely,
Q
is decomposed as
Q
(
θ
)
=
Q
N
−
1
(θ
N
−
1
)
Q
N
−
2
(θ
N
−
2
)
···
Q
1
(θ
1
)
T
, with
θ
i
∈]−
where
θ
=[
θ
1
,θ
2
,...,θ
N
−
1
]
π/
2
,π/
2
[
,1
≤
i
≤
(N
−
1
)
, and
Q
i
(θ
i
)
=
Q
i,N
(θ
i
)
.Matrix
Q
can be further split into two terms:
=
Q
(
θ
)
q
(
θ
)
Q
(
θ
)
in which
Q
(
θ
)
N
×
(N
−
1
)
and
∈ R
q
(
θ
)
=[
sin
θ
1
,
cos
θ
1
sin
θ
2
,...,
T
N
cos
θ
1
···
cos
θ
N
−
2
sin
θ
N
−
1
,
cos
θ
1
···
cos
θ
N
−
2
cos
θ
N
−
1
]
∈ R
represents the extracting vector for the source currently targeted as
q
T
z
.
y
=
(9.16)
Angular parameters
θ
are estimated through a gradient update, much like those sum-
marized in the previous sections. More importantly, by the structure of the mixing
matrix after prewhitening, the extracting vector
q
(
θ
)
lies orthogonal to all columns
of matrix
Q
(
θ
)
,
Q
T
(
θ
)
z
1
is uncorrelated with
y
, the source extracted by
q
(
θ
)
. Hence, to extract the next source, the algorithm can
be repeated using
N
−
∀
θ
. As a result, the vector
˜
z
∈ R
z
instead of
z
and reducing the dimensions of
θ
accordingly. The
uncorrelation of
z
and
y
prevents the same source from being extracted again. This
dimensionality reduction, achieved by the particular parametrization of the orthog-
onal mixing matrix after prewhitening in the real-valued case, reduces the computa-
tional cost after each deflation stage.
˜
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