Digital Signal Processing Reference
In-Depth Information
An important aspect is related to the ambiguities of the solution. As
in the instantaneous case, separation of convolutive mixtures also exhibits
a permutation indeterminacy, which is only solved if additional informa-
tion about the sources is provided. Nonetheless, the amplitude ambiguity is
now replaced by a filtering indeterminacy. This occurs because if
y
1
(
n
)
and
ˆ
y
2
(
n
)
are mutually independent processes, so will be
y
1
(
n
)
=
h
1
(
n
)
∗
y
1
(
n
)
ˆ
and
. There-
fore, unless additional information about the original sources is available,
the method is susceptible to providing distorted versions of the sources.
A number of different methods based on a time-domain approach can
be found in the literature. An extension of Hérault and Jutten's algorithm
is proposed in [159], in which the coefficients of the separating structure,
as shown in Figure 6.5, are replaced by linear filters. Other methods have
been modified to cope with the convolutive mixture model, like the natural
gradient algorithm [10,11], Infomax [290,291], as well as the FastICA [285].
y
2
(
n
)
=
h
2
(
n
)
∗
y
2
(
n
)
, for any invertible filters
h
1
(
n
)
and
h
2
(
n
)
6.5.2 Signal Separation in the Frequency Domain
Another class of methods for the case of convolutive mixtures deals with
separation in the frequency domain [198,276].
The observed signals
x
i
(
n
)
, sampled at a frequency
f
s
, can be represented
in the frequency domain
by means of a short-time Fourier transform
(STFT) with a finite number of points, i.e.,
x
i
(
¯
f
, τ
)
L
2
+
1
x
i
f
, τ
=
x
i
τ
l
w
window
l
e
−
j
2π
fl
¯
+
(6.96)
L
2
l
=−
where
f
0,
L
f
s
,
,
L
−
L
f
s
denotes the set of frequency bins
∈
...
w
window
(
is a windowing function
τ represents a time index
n
)
Therefore, the observed signal can be approximately represented as the
result of a linear instantaneous mixture in a frequency bin, i.e.,
x
i
f
, τ
=
a
ij
f
¯
s
j
f
, τ
¯
¯
(6.97)
j
where
¯
a
ij
(
n
)
is the frequency response between the
j
th source and the
i
th sensor
¯
s
j
(
f
, τ
)
denotes the STFT of the
j
th source
Finally, organizing (6.97) into matrix notation we get to
x
f
, τ
=
A
f
¯
s
f
, τ
¯
(6.98)
