04-04-2012, 03:08 PM
Blind Source Separation (BSS)
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INTRODUCTION
In the field of digital signal processing there is a problem known as Cocktail Party, which try to separate signals (voice or music) mixed simultaneously based only on their mixtures.Blind Source Separation (BSS) is a powerful technique capable of solving this problem.BSS have applications in mobile telephony, multiuser communication systems, eliminating redundancy and sparse coding in noise cancellation, voice reinforcement in noisy environments, as well as in other important environments such as urban ecology specifically on pollution caused by high sound levels. This technique is based on the following principle: assuming that the original signals are mixed linearly and it is possible to collect these mixtures with appropriate sensors, the BSS is able to estimate the coefficients that characterize this linear combination, and therefore can estimate the original signals.
INDEPENDENT COMPONENT ANALYSIS (ICA)
The Independent Components Analysis algorithm allows two source signals to be separated from two linear mixtures of the source signals using statistical principles of independence and nongaussianity.ICA assumes that the value of each source at any given time is a random variable.It also assumes that each source is statistically independent. This implies that the values of one source cannot be correlated to values in any of the other sources.With these assumptions, ICA allows us to separate source signals from mixtures of these source signals.The algorithm requires that there be as many sensors as input signals.It is based on higher order statistics like Kurtosis, Negentropy etc where complete statistical independence of the signals is the primary concern.
1.2 PRINCIPAL COMPONENT ANALYSIS (PCA)
Principal component analysis is a technique that is useful for the compression and classification of data.It involves a mathematical procedure that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables called principal components.The main purpose is to reduce the dimensionality of a data by finding a new set of variables retaining most of the information.PCA requires the calculation of the eigen value decomposition of a data covariance matrix or singular value decomposition of a data matrix, usually after mean centering the data.Analysis is done in the time domain.
1.3 BEAMFORMING
Beamforming is the process of trying to concentrate the array to sounds coming from only one particular direction.Beamformer is a spatial filter that process data obtained from an array and tries to extract the desired signal from the background noise and interferences. Sensor array collects spatial samples of propagating signal source.Parameters in beamforming are adjusted to form a spatial pattern with a dominant response for the direction of interest while the response for the position of interfering signals is minimized.
MOTIVATION
ICA or Independent Component Analysis is a hugely researched technique for blind source separation.ICA has developed a lot over the years and several algorithms and methods have been researched till date.The problem is to separate two or more signals which have been linearly combined to generate mixed signals which are available to us but we have no prior knowledge about the source signals.A possible real life situation where ICA can be used is to separate the voice from the noise while using a mobile phone when the noise is too high.
2.2 ICA APPROACH
The starting point for ICA is the very simple assumption that the components the sources s_i are statistically independent.A “source” means here an original signal i.e. independent component,like the speaker in a cocktail party problem. “Blind” means that we have no or very little prior knowledge about the mixing matrix, and make very few assumptions about the source signals.
2.3 PRE-PROCESSING FOR ICA
Pre-processing step in ICA is to make the sure that the observed mixed signals have zero mean, unit variance and are de-correlated. The de-correlation removes the second-order dependencies between the observed signals. The following methods are used to pre-process the ICA data.
WHY GAUSSIAN VARIABLES ARE FORBIDDEN?
The fundamental restriction in ICA is that the independent components must be non-Gaussian for ICA algorithms to work. Gaussian variables make ICA impossible. The figure below shows the joint distribution of two uncorrelated Gaussian variables of unit variance. Clearly, it does not contain any information on the directions of the columns of the mixing matrix A. This is why A cannot be estimated from the above data.
AMBIGUITIES AND LIMITATIONS
The following significant ambiguities arise in the ICA algorithm.
3.1 INDETERMINATE ENERGY:
Because a scalar multiplier could be pulled out of s and multiplied to A with no change in the above equations, the ICA algorithm cannot determine the energy contained in any of the independent sources it finds. The amplitudes it gives the output components are arbitrary, and the true source signal could be one the isolated sources multiplied by any scalar multiple. This includes a negative multiple, which means that often, the output signals are also inversions of the original signals. This may not be a problem with speech signals but may be a problem with images where energy will matter.
3.2 ORDER AMBIGUITY:
Because the algorithm chooses coefficients of “W” at random when it searches for the sources, the isolated sources that the algorithm finds can come out in any order. So, it would take some additional processing to determine which independent sources is the one of interest to us.
3.3 UNDER-DETERMINATION:
There must be as many sensors as there are sources in order to properly isolate the sources. If there are not enough sensors, the resulting signals will not match any of the sources, but rather will still be mixtures of multiple sources.
3.4 LINEARITY:
ICA can only handle linear mixtures that can be represented in the form x = As. The algorithm cannot accurately guess the independent sources if the sources are out of phase in the mixtures or if the mixtures have other nonlinear features.
CONCLUSION
The BSS technique is used to achieve signal separation in circumstances where the only information available is the received signals or the observed signals whose mixing process remains unknown. It also does not have any knowledge about the characteristics of signals and the source from which the signals have originated. BSS can be applied to a variety of situations such as, the separation of simultaneous speakers, analysis of biomedical signals obtained by EEG or in wireless telecommunications to separate several received signals.We have to recover the independent source signals given only the sensor readings composed of unknown linear combinations of the independent sources. We can successfully separate the signals or separate the signals from the background noise.
[attachment=19466]
INTRODUCTION
In the field of digital signal processing there is a problem known as Cocktail Party, which try to separate signals (voice or music) mixed simultaneously based only on their mixtures.Blind Source Separation (BSS) is a powerful technique capable of solving this problem.BSS have applications in mobile telephony, multiuser communication systems, eliminating redundancy and sparse coding in noise cancellation, voice reinforcement in noisy environments, as well as in other important environments such as urban ecology specifically on pollution caused by high sound levels. This technique is based on the following principle: assuming that the original signals are mixed linearly and it is possible to collect these mixtures with appropriate sensors, the BSS is able to estimate the coefficients that characterize this linear combination, and therefore can estimate the original signals.
INDEPENDENT COMPONENT ANALYSIS (ICA)
The Independent Components Analysis algorithm allows two source signals to be separated from two linear mixtures of the source signals using statistical principles of independence and nongaussianity.ICA assumes that the value of each source at any given time is a random variable.It also assumes that each source is statistically independent. This implies that the values of one source cannot be correlated to values in any of the other sources.With these assumptions, ICA allows us to separate source signals from mixtures of these source signals.The algorithm requires that there be as many sensors as input signals.It is based on higher order statistics like Kurtosis, Negentropy etc where complete statistical independence of the signals is the primary concern.
1.2 PRINCIPAL COMPONENT ANALYSIS (PCA)
Principal component analysis is a technique that is useful for the compression and classification of data.It involves a mathematical procedure that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables called principal components.The main purpose is to reduce the dimensionality of a data by finding a new set of variables retaining most of the information.PCA requires the calculation of the eigen value decomposition of a data covariance matrix or singular value decomposition of a data matrix, usually after mean centering the data.Analysis is done in the time domain.
1.3 BEAMFORMING
Beamforming is the process of trying to concentrate the array to sounds coming from only one particular direction.Beamformer is a spatial filter that process data obtained from an array and tries to extract the desired signal from the background noise and interferences. Sensor array collects spatial samples of propagating signal source.Parameters in beamforming are adjusted to form a spatial pattern with a dominant response for the direction of interest while the response for the position of interfering signals is minimized.
MOTIVATION
ICA or Independent Component Analysis is a hugely researched technique for blind source separation.ICA has developed a lot over the years and several algorithms and methods have been researched till date.The problem is to separate two or more signals which have been linearly combined to generate mixed signals which are available to us but we have no prior knowledge about the source signals.A possible real life situation where ICA can be used is to separate the voice from the noise while using a mobile phone when the noise is too high.
2.2 ICA APPROACH
The starting point for ICA is the very simple assumption that the components the sources s_i are statistically independent.A “source” means here an original signal i.e. independent component,like the speaker in a cocktail party problem. “Blind” means that we have no or very little prior knowledge about the mixing matrix, and make very few assumptions about the source signals.
2.3 PRE-PROCESSING FOR ICA
Pre-processing step in ICA is to make the sure that the observed mixed signals have zero mean, unit variance and are de-correlated. The de-correlation removes the second-order dependencies between the observed signals. The following methods are used to pre-process the ICA data.
WHY GAUSSIAN VARIABLES ARE FORBIDDEN?
The fundamental restriction in ICA is that the independent components must be non-Gaussian for ICA algorithms to work. Gaussian variables make ICA impossible. The figure below shows the joint distribution of two uncorrelated Gaussian variables of unit variance. Clearly, it does not contain any information on the directions of the columns of the mixing matrix A. This is why A cannot be estimated from the above data.
AMBIGUITIES AND LIMITATIONS
The following significant ambiguities arise in the ICA algorithm.
3.1 INDETERMINATE ENERGY:
Because a scalar multiplier could be pulled out of s and multiplied to A with no change in the above equations, the ICA algorithm cannot determine the energy contained in any of the independent sources it finds. The amplitudes it gives the output components are arbitrary, and the true source signal could be one the isolated sources multiplied by any scalar multiple. This includes a negative multiple, which means that often, the output signals are also inversions of the original signals. This may not be a problem with speech signals but may be a problem with images where energy will matter.
3.2 ORDER AMBIGUITY:
Because the algorithm chooses coefficients of “W” at random when it searches for the sources, the isolated sources that the algorithm finds can come out in any order. So, it would take some additional processing to determine which independent sources is the one of interest to us.
3.3 UNDER-DETERMINATION:
There must be as many sensors as there are sources in order to properly isolate the sources. If there are not enough sensors, the resulting signals will not match any of the sources, but rather will still be mixtures of multiple sources.
3.4 LINEARITY:
ICA can only handle linear mixtures that can be represented in the form x = As. The algorithm cannot accurately guess the independent sources if the sources are out of phase in the mixtures or if the mixtures have other nonlinear features.
CONCLUSION
The BSS technique is used to achieve signal separation in circumstances where the only information available is the received signals or the observed signals whose mixing process remains unknown. It also does not have any knowledge about the characteristics of signals and the source from which the signals have originated. BSS can be applied to a variety of situations such as, the separation of simultaneous speakers, analysis of biomedical signals obtained by EEG or in wireless telecommunications to separate several received signals.We have to recover the independent source signals given only the sensor readings composed of unknown linear combinations of the independent sources. We can successfully separate the signals or separate the signals from the background noise.