Linear Instantaneous TIme-Frequency Ratio Of Mixtures

LI-TIFROM is a sparsity-based Blind Source Separation (BSS) method. It is based on a time-frequency (TF) analysis :

• it first finds single-source TF zones, i.e. a set of adjacent TF windows, where a single source is active.
• Then, in each of the above zones, it estimates a column of the mixing matrix.
• When all the columns of the mixing matrix have been estimated, the last step consists in recovering the sources.

Keywords : blind source separation ; sparsity ; time-frequency analysis ; stationary, non-stationary and/or dependent sources ;

LI-TIFROM est une méthode de Séparation Aveugle de Sources (SAS). Elle est fondée sur une analyse temps-fréquence (TF) :

• elle trouve d’abord des zones TF mono-sources, c.-à-d. un ensemble de fenêtres TF adjacentes, où une source est seule à être active.
• Ensuite, dans chacune de ces zones, elle estime une colonne de la matrice de mélange.
• Quand toutes les colonnes de la matrice de mélange ont été estimées, la dernière étape consiste à reconstruire les sources.

Mots clés : séparation aveugle de sources ; parcimonie ; analyse temps-fréquence ; sources stationnaires, non-stationnaires et/ou dépendantes ;

Please acknowledge the use of this software in any publication : "The LI-TIFROM software is available at http://www.ast.obs-mip.fr/bss-softwares" and cite the references :

• F. Abrard and Y. Deville, A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources, Signal Processing, vol. 85, issue 7, pp. 1389-1403, July 2005.
• Y. Deville, M. Puigt, and B. Albouy, Time-frequency blind signal separation : extended methods, performance evaluation for speech sources, Proceedings of the IEEE International Joint Conference on Neural Networks (IJCNN 2004), pp. 255-260, Budapest, Hungary, July 25-29, 2004.
• M. Puigt and Y. Deville, Time-frequency ratio-based blind separation methods for attenuated and time-delayed sources, Section 3 : "Summary of the TIFROM method for linear instantaneous mixtures", Mechanical Systems and Signal Processing, vol. 19, issue 6, pp. 1348-1379, November 2005.

Please send a copy of your publication to Matthieu.Puigt@ast.obs-mip.fr and Yannick.Deville@ast.obs-mip.fr .

Contents

• 1 Overview
• 2 How to use LI-TIFROM ?
  • 2.1 Download
  • 2.2 Versions of LI-TIFROM
  • 2.3 How to use it ?
  • 2.4 Input parameters options

  
  
• 3 Examples
  • 3.1 Determined mixtures
  • 3.2 Underdetermined mixtures
  
  
• 4 Contact us

1 Overview

Blind source separation (BSS) consists in estimating a set of unknown sources from a set of observations resulting from mixtures of these sources through unknown propagation channels. The mixing model that we consider here is the linear instantaneous mixture :

Unlike the majority of the proposed methods (based on such tools as higher-order statistics or information theory), some approaches based on time-frequency representations appeared since the last decade. Some of these methods suppose that at most one source is active in each atom of the analysis domain. On the contrary, the LI-TIFROM method only needs one tiny single-source time-frequency zone per source in order to estimate the mixing matrix, while allowing several sources to be active in all other zones.

This approach is based on the following stages :

1. it first computes Short-Time Fourier Transform (STFT) of the sources.
2. Detection stage : it finds Time-Frequency (TF) zones where a single source is active (called single-source zones).
3. Identification stage : it estimates a column of the unknown mixing matrix in each of the above zones and only keeps the "best" "distant" columns (in the sense of the single-source quality measured in Step 2).
4. Reconstruction stage : it eventually recovers the sources.

Two versions have been proposed (See Subsection 2.2 for more explanations) and details can be found in :

• F. Abrard and Y. Deville, A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources, Signal Processing, vol. 85, issue 7, pp. 1389-1403, July 2005.
• Y. Deville, M. Puigt, and B. Albouy, Time-frequency blind signal separation : extended methods, performance evaluation for speech sources, Proceedings of the IEEE International Joint Conference on Neural Networks (IJCNN 2004), pp. 255-260, Budapest, Hungary, July 25-29, 2004.
• M. Puigt and Y. Deville, Time-frequency ratio-based blind separation methods for attenuated and time-delayed sources, Section 3 : "Summary of the TIFROM method for linear instantaneous mixtures", Mechanical Systems and Signal Processing, vol. 19, issue 6, pp. 1348-1379, November 2005.

2 How to use LI-TIFROM ?

As numerous TF-BSS approaches, the LI-TIFROM

method needs to select "good" parameters in order to achieve the separation ! In particular, depending on your data and the selected parameters, it still may fail or provide inaccurate results. In this case, please, do not give up ! Send your comments to Matthieu.Puigt@ast.obs-mip.fr and Yannick.Deville@ast.obs-mip.fr. We will do our best to help you.

2.1 Download

The LI-TIFROM package is written in MATLAB , needs the MATLAB Signal Processing Toolbox in order to run and has been tested on MATLAB 6.0, 7.0 and 7.5. This program is under the CeCILL license so you can download the sources and modify them under the terms of this license.

2.2 Versions of LI-TIFROM

The first version of LI-TIFROM has been proposed at the beginning of the 2000’s in :

• F. Abrard and Y. Deville, A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources, Signal Processing, vol. 85, issue 7, pp. 1389-1403, July 2005.

However, we showed that the variance of the ratio of observations in the above LI-TIFROM approach has an asymmetrical behaviour (see e.g. Y. Deville, M. Puigt, Temporal and time-frequency correlation-based blind source separation methods. Part I : Determined and underdetermined linear instantaneous mixtures, Signal Processing, vol. 87, issue 3, pp. 374-407, March 2007). As a consequence, we proposed a Symmetrical version, hence its name LI-TIFROM-S, which takes into account the ratio and its inverse .

Moreover, in the case when the number of observations is higher than , we showed that it was better to study all ratios

and

and study the variance of the means over of these ratios to find single-source zones.

All these improvements have been proposed and detailed in :

• Y. Deville, M. Puigt, and B. Albouy, Time-frequency blind signal separation : extended methods, performance evaluation for speech sources, Proceedings of the IEEE International Joint Conference on Neural Networks (IJCNN 2004), pp. 255-260, Budapest, Hungary, July 25-29, 2004.
• M. Puigt and Y. Deville, Time-frequency ratio-based blind separation methods for attenuated and time-delayed sources, Section 3 : "Summary of the TIFROM method for linear instantaneous mixtures", Mechanical Systems and Signal Processing, vol. 19, issue 6, pp. 1348-1379, November 2005.

Both versions of LI-TIFROM (i.e. asymmetrical and symmetrical ones) are available in this package.

2.3 How to use it ?

In the proposed package, as explained above, two versions of

LI-TIFROM that we resp. named li_tifrom and li_tifrom_sym are provided. Both methods can estimate the unknown mixing matrix in the determined () and underdetermined () case. They use the same parameters :

[B,coord,compt,Ns_found,error,X,f,t] = ...
li_tifrom(x,N_sources,nb_samp_in_win,overlap,nb_win,fs) ;

[B,coord,compt,Ns_found,error,X,f,t] = ...
li_tifrom_sym(x,N_sources,nb_samp_in_win,overlap,nb_win,fs) ;

1. Input parameters :
   • x : the 2D matrix of observations, where each row is an observation.
   • N_sources : the number of unknown sources.
   • nb_samp_in_win : the number of samples in temporal windows in STFT computations. By default, this value is set to 128.
   • overlap : the temporal overlap between two time-adjacent TF windows. By default, this value is set to 0.75.
   • nb_win : the number of TF windows in TF zones. By default, this value is set to 10.

   • fs : the sampling frequency. By default, this value is set to 1.

2. Output parameters :
   • B : the estimated mixing matrix.
   • coord : the 2D matrix of the coordinates of the TF zones used to find the columns of B.
   • compt : the number of TF zones used to estimate B.

   • Ns_found : the number of sources which have been found.
   • error : the parameter which indicates why the program stops. The possible values of error are 0,1 and 2 :
     • if , then all the columns of the mixing matrix have been estimated.
     • if , then at least one column of the mixing matrix has not been estimated.In particular, the program stopped because the variance of the ratios is above a user-defined threshold corresponding to the maximum authorized variance for a TF zone to be "really" single-source. It allows us to speed up the computations, by occulting some TF analysis zones. In our first papers, the value of this threshold was set to 1.5e-2. We now omit it by putting a high value !
     • if , then all the TF zones have been studied but at least one column of the mixing matrix has not been found.

   • X : the 3D matrix of TF observations (i.e. STFTs of observations).
   • f : the vector of frequencies at which the STFTs are computed.
   • t : the vector of time samples at which the STFTs are computed.

2.4 Input parameters options

For both methods, some input parameters are optional, as we can see below :

• fs is only needed if you want to plot STFTs of observations, using output parameters X, f and

  t. When it is not selected, e.g.

  [B,coord,compt,Ns_found,error,X,f,t] = ...
  li_tifrom(x,N_sources,nb_samp_in_win,overlap,nb_win) ;
  

  fs is set to 1.
• nb_win is the number of TF windows per analysis zone. In our tests, we found that, for the tested signals, it may be set to 10. In the case below,

  [B,coord,compt,Ns_found,error,X,f,t] = ...
  li_tifrom(x,N_sources,nb_samp_in_win,overlap) ;
  

  fs is set to 1 and nb_win is set to 10.

• overlap is the temporal overlap between two TF windows in STFTs computations. In our tests, we found that, for the tested signals, it may be set to 75%. With the following script,

  [B,coord,compt,Ns_found,error,X,f,t] = ...
  li_tifrom(x,N_sources,nb_samp_in_win) ;
  

  fs, nb_win and overlap are resp. set to 1, 10 and 0.75.
• Lastly, nb_samp_in_win is the number of samples per TF window in STFTs computations. In our tests with speech signals sampled at 20kHz, we showed that this value should be preferably set to 128 (or 256).

  [B,coord,compt,Ns_found,error,X,f,t] = li_tifrom(x,N_sources) ;
  

  In this case, fs, nb_win, overlap and nb_samp_in_win are resp. set to 1, 10, 0.75 and 128.

3 Examples

In this section, we propose 4 examples provided in the LI-TIFROM package, involving several configurations. The first examples concern the determined mixtures, i.e. when the number of sources is equal to the number of observations. The example in Subsection 3.2

illustrates the performance of the proposed approaches when there are more sources than observations ().

The sources are real English spoken signals, from the MULTEXT database¹. All these signals have first been centered and normalized, so that their higher absolute value is equal to 1. Temporal and TF representations of these signals are presented in Figure 1.

Figure 1 : Temporal and TF representations of the sources.

Depending whether the mixtures are determined or underdetermined, we use different measure performance criteria :

1. In all the examples of Subsection 3.1, we compute the Signal-To-Interference Ratio () parameters² :
   • the input SIR () defines the "mixture rate" : the more the sources are mixed, the lower of the values of this parameter take.
   • The output SIR () defines the "separation quality".

   • The SIR Improvement () is the improvement provided by the BSS methods :
     

     
     

     
     

     

     
     

2. In the underdetermined case presented in Subsection 3.2, recovering the sources by inverting the mixing matrix is not possible. What has been proposed by our team consists in considering an invertible submatrix of the estimated mixing matrix and in separating the sources partially³. In the example provided in Subsection 3.2, we propose a global measure criterion by computing the Frobenius norm of the difference between theoretical and estimated mixing matrices.

3.1 Determined mixtures

In this subsection, we propose three examples :

1. demo1 shows the performance of the proposed approaches when with a "strong" mixture, i.e. we mix real speech signals with the following mixing matrix :
   

   

   
   

   
   Performance achieved by the LI-TIFROM methods :
   SIR^in : 1.4 dB
   
   "Asymmetrical" LI-TIFROM method :
   SIR^out : 82.0 dB
   SIRI : 80.6 dB
   
   "Symmetrical" LI-TIFROM method :
   SIR^out : 82.0 dB
   SIRI : 80.6 dB
   

2. demo2 shows the performance of the proposed approaches when with a "weak" mixture, i.e. we mix the same signals with the following mixing matrix :
   

   

   
   

   This case illustrates the improvement provided by the symmetrical version of LI-TIFROM :

   
   Performance achieved by the LI-TIFROM methods :
   SIR^in : 30.0 dB
   
   "Asymmetrical" LI-TIFROM method :
   SIR^out : -34.4 dB
   SIRI : -64.4 dB
   
   "Symmetrical" LI-TIFROM method :
   SIR^out : 68.3 dB
   SIRI : 38.3 dB
   

3. demo3 shows the performance of the proposed approches when (the sparsity in the observations is reduced when sources are numerous). The mixing matrix is set to :
   

   

   
   

   As expected, when the number of observations is above 2, the LI-TIFROM-S approach yields better performance :

   
   Performance achieved by the LI-TIFROM methods :
   SIR^in : -3.6 dB
   
   "Asymmetrical" LI-TIFROM method :
   SIR^out : -64.6 dB
   SIRI : -61.0 dB
   
   "Symmetrical" LI-TIFROM method :
   SIR^out : 43.0 dB
   SIRI : 46.7 dB
   

3.2 Underdetermined mixtures

Contrary to numerous classical BSS approaches, LI-TIFROM(-S) deals with underdetermined mixtures, while taking advantage of the sparsity assumption. We here show the performance of the approaches in this case. In demo4, we mix sources in observations with the following matrix :

Performance achieved by the LI-TIFROM methods :

"Asymmetrical" LI-TIFROM method :
Frobenius norm : 1.6e-003

"Symmetrical" LI-TIFROM method :
Frobenius norm : 1.6e-003

4 Contact us

The LI-TIFROM package is currenlty maintained by Matthieu PUIGT and Yannick DEVILLE. Many thanks to Frédéric ABRARD who developped the first version of this software.

5 About this document ...

This document was generated using the LaTeX2HTML translator Version 2002-2-1 (1.71)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

Footnotes

... database¹

http://aune.lpl.univ-aix.fr/projects/MULTEXT/

... parameters²

These parameters are e.g. defined in : Y. Deville, M. Puigt, Temporal and time-frequency correlation-based blind source separation methods. Part I : Determined and underdetermined linear instantaneous mixtures, Signal Processing, vol. 87, issue 3, pp. 374-407, March 2007.

... partially³

See : F. Abrard and Y. Deville, A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources, Signal Processing, vol. 85, issue 7, pp. 1389-1403, July 2005.