Digital Signal Processing Reference
In-Depth Information
cil
(
R
s1
.
R
s2
) formed with the source signals and the matrix pencil
(
R
x1
,
R
x2
) formed with the sensor signals. Both pencils are considered
congruent
if there exists an invertible matrix
A
∈
Gl(
n
) such that
AR
s1
A
T
R
x1
=
AR
s2
A
T
.
R
x2
=
(14.1)
,i
=1
, ..., m, j
=1
, ..., n
represents
the instantaneous mixing matrix. It has been shown that the inverse or
pseudo inverse of the mixing matrix can be estimated from the sensor
signal pencil if the eigenvector matrix of the source signal pencil is
diagonal. In fact, congruent pencils possess the same eigenvalues which
form the roots of the characteristic polynomials
In BSS problems
A
=
{
a
ij
}
χ
x
(
λ
)=d
R
x1
−
R
x2
)=0
χ
s
(
λ
)=d
R
s1
−
R
s2
) = 0
(14.2)
With
A
a rectangular matrix (
m>n
), if
A
T
A
is an invertible ma-
trix, the congruent source signal pencil (
A
T
AR
s1
A
T
A
,
A
T
AR
s2
A
T
A
)
also possesses the same eigenvalues. Hence the sensor signal pencil
formed with (
m
m
) matrices shows
n
eigenvalues equal to the eigen-
values of the source signal pencil.
The generalized eigendecomposition of the sensor signal pencil now
reads
×
R
x1
E
=
R
x2
EΛ
(14.3)
where
E
represents the unique eigenvector matrix if the diagonal matrix
Λ
has distinct eigenvalues
λ
i
. The corresponding eigendecomposition
statement concerning the source signal pencil can be obtained easily by
substituting equation(14.1) into equation(14.3), yielding
AR
s1
A
T
E
=
AR
s2
A
T
EΛ
(14.4)
Multiplying both sides of equation(14.4) by
A
−1
and using
E
s
=
A
T
E
(14.5)
as the corresponding eigendecomposition statement of the source signal
pencil results in
R
s1
E
s
=
R
s2
E
s
Λ
,
(14.6)
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