Digital Signal Processing Reference
In-Depth Information
Fig. 9.4
Schematic diagram of the system setup and notations used by algorithms based on refer-
ence signals
respect, the resulting method can be considered as
semi-blind
[
12
]. The above nota-
tions are graphically summarized in Fig.
9.4
.
Now, given a reference signal
z(n)
we can define the following criterion:
|
C
z
{
y
}|
J
r
(
w
,
v
)
(9.38)
E
{|
y(n)
|
2
}
E
{|
z(n)
|
2
}
where
Cum
y(n),y(n)
∗
,z(n),z(n)
∗
is a fourth-order cross-cumulant defined for any jointly stationary signals
z(n)
and
y(n)
. The first interesting property of criterion (
9.38
) is that it is a contrast
function for almost any fixed reference signal
z(n)
. Precise conditions for the va-
lidity of such a contrast are derived in [
10
,
12
,
16
]. Essentially, the reference
z(n)
should be 'close' enough to one particular source signal so as not to contain identical
power contributions from two (or more) sources. This assumption is generally sat-
isfied in practice even if the reference filter is chosen randomly. More interestingly,
function (
9.38
) can be expressed as
C
z
{
y
}
w
H
C
v
w
|
(
w
H
Rw
)(
v
H
Rv
)
.
|
J
r
(
w
,
v
)
=
(9.39)
In the above equation,
R
and
C
v
are a covariance and a cumulant matrix which, in
the case of an instantaneous mixture, are given by:
E
x
(n)
x
(n)
H
,
Cum
x
i
(n),x
j
(n)
∗
,z(n),z(n)
∗
.
R
=
[
C
v
]
ij
=
Replacing
x
(n)
by a vector stacking the consecutive delayed values of the observa-
tion as in Sect.
9.2.2
, a similar definition holds in the convolutive case (see [
13
,
16
]
for details).
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