Electrical audio signal processing systems and devices – Noise or distortion suppression
Reexamination Certificate
1997-07-11
2001-02-06
Isen, Forester W. (Department: 2747)
Electrical audio signal processing systems and devices
Noise or distortion suppression
C381S066000, C381S094200, C381S071100
Reexamination Certificate
active
06185309
ABSTRACT:
BACKGROUND OF THE INVENTION
The present invention relates generally to separating individual source signals from a mixture of the source signals and more specifically to a method and apparatus for separating convolutive mixtures of source signals.
A classic problem in signal processing, best known as blind source separation, involves recovering individual source signals from a mixture of those individual signals. The separation is termed ‘blind’ because it must be achieved without any information about the sources, apart from their statistical independence. Given L independent signal sources (e.g., different speakers in a room) emitting signals that propagate in a medium, and L′ sensors (e.g., microphones at several locations), each sensor will receive a mixture of the source signals. The task, therefore, is to recover the original source signals from the observed sensor signals. The human auditory system, for example, performs this task for L′=2. This case is often referred to as the ‘cocktail party’ effect; a person at a cocktail party must distinguish between the voice signals of two or more individuals speaking simultaneously.
In the simplest case of the blind source separation problem, there are as many sensors as signal sources (L=L′) and the mixing process is instantaneous, i.e., involves no delays or frequency distortion. In this case, a separating transformation is sought that, when applied to the sensor signals, will produce a new set of signals which are the original source signals up to normalization and an order permutation, and thus statistically independent. In mathematical notation, the situation is represented by
u
^
i
(
t
)
=
∑
j
L
g
ij
v
j
(
t
)
(
1
)
where g is the separating matrix to be found, v(t) are the sensor signals and u(t) are the new set of signals.
Significant progress has been made in the simple case where L=L′ and the mixing is instantaneous. One such method, termed independent component analysis (ICA), imposes the independence of u(t) as a condition. That is, g should be chosen such that the resulting signals have vanishing equal-time cross-cumulants. Expressed in moments, this condition requires that
<
û
i
(
t
)
m
û
j
(
t
)
n
>=<û
i
(
t
)
m
><û
j
(
t
)
n
>
for i=j and any powers m, n; the average taken over time t. However, equal-time cumulant-based algorithms such as ICA fail to separate some instantaneous mixtures such as some mixtures of colored Gaussian signals, for instance.
The mixing in realistic situations is generally not instantaneous as in the above simplified case. Propagation delays cause a given source signal to reach different sensors at different times. Also, multi-path propagation due to reflection or medium properties creates multiple echoes, so that several delayed and attenuated versions of each signal arrive at each sensor. Further, the signals are distorted by the frequency response of the propagation medium and of the sensors. The resulting ‘convolutive’ mixtures cannot be separated by ICA methods.
Existing ICA algorithms can separate only instantaneous mixtures. These algorithms identify a separating transformation by requiring equal-time cross-cumulants up to arbitrarily high orders to vanish. It is the lack of use of non-equal-time information that prevents these algorithms from separating convolutive mixtures and even some instantaneous mixtures.
As can be seen from the above, there is need in the art for an efficient and effective learning algorithm for blind separation of convolutive, as well as instantaneous, mixtures of source signals.
SUMMARY OF THE INVENTION
In contrast to existing separation techniques, the present invention provides an efficient and effective signal separation technique that separates mixtures of delayed and filtered source signals as well as instantaneous mixtures of source signals inseparable by previous algorithms. The present invention further provides a technique that performs partial separation of source signals where there are more sources than sensors.
The present invention provides a novel unsupervised learning algorithm for blind separation of instantaneous mixtures as well as linear and non-linear convoluted mixtures, termed Dynamic Component Analysis (DCA). In contrast with the instantaneous case, convoluted mixtures require a separating transformation g
ij
(t) which is dynamic (time-dependent): because a sensor signal v
i
(t) at the present time t consists not only of the sources at time t but also at the preceding time block t−T≦t′<t of length T, recovering the sources must, in turn, be done using both present and past sensor signals, v
i
(t′≦t). Hence:
u
^
i
(
t
)
=
∑
j
=
0
L
∫
0
∞
ⅆ
t
′
g
ij
(
t
′
)
v
j
(
t
-
t
′
)
(
2
)
The simple time dependence g
ij
(t)=g
ij
&dgr;(t) reduces the convolutive to the instantaneous case. In general, the dynamic transformation g
ij
(t) has a non-trivial time dependence as it couples mixing with filtering. The new signals u
i
(t) are termed the dynamic components (DC) of the observed data; if the actual mixing process is indeed linear and square (i.e., where the number of sensors L′ equals the number of signal sources L), the DCs correspond to the original sources.
To find the separating transformation g
ij
(t) of the DCA procedure, it first must be observed that the condition of vanishing equal time cross-cumulance described above is not sufficient to identify the separating transformation because this condition involves a single time point. However, the stronger condition of vanishing two-time cross-cumulants can be imposed by invoking statistical independence of the sources, i.e.,
<û
i
(t)
m
û
j
(t+&tgr;)
n
>=<û
i
(t)
m
><û
j
(t+&tgr;)
n
>,
for i≠j in any powers m, n at any time &tgr;. This is because the amplitude of source i at time t is independent of the amplitude of source j≠i at any time t+&tgr;. This condition requires processing the sensor signals in time blocks and thus facilitates the use of their temporal statistics to deduce the separating transformation, in addition to their intersensor statistics.
An effective way to impose the condition of vanishing two-time cross-cumulants is to use a latent variable model. The separation of convoluted mixtures can be formulated as an optimization problem: the observed sensor signals are fitted to a model of mixed independent sources, and a separating transformation is obtained from the optimal values of the model parameters. Specifically, a parametric model is constructed for the joint distribution of the sensor signals over N-point time blocks, p
v
[v
1
(t
1
) . . . , v
1
(t
N
) , . . . , v
L′
(t
1
), . . . , v
L′
(t
N
)]. To define p
v
, the sources are modeled as independent stochastic processes (rather than stochastic variables), and a parameterized model is used for the mixing process which allows for delays, multiple echoes and linear filtering. The parameters are then optimized iteratively to minimize the information-theory distance (i.e., the Kullback-Leibler distance) between the model sensor distribution and the observed distribution. The optimized parameter values provide an estimate of the mixing process, from which the separating transformation g
ij
(t) is readily available as its inverse.
Rather than work in the time domain, it is technically convenient to work in the frequency domain since the model source distribution factorizes there. Therefore, it is convenient to preprocess the signals using Fourier transform and to work with the Fourier components V
i
(w
k
).
In the linear version of DCA, the only information about the sensor signals used by the estimation procedure is their cross-correlations <v
i
(t)v
j
(t′)> (or, equivalently, their cross-spectra <V
i
(w)V
j
*(w)>). This provides a computational advantage, leading to simple learning rules and fast convergence. Another a
Isen Forester W.
Mei Xu
The Regents of the University of California
Townsend and Townsend / and Crew LLP
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