Docstoc

BSS Reju

Document Sample
BSS Reju Powered By Docstoc
					  Blind Separation of Speech Mixtures


             Vaninirappuputhenpurayil Gopalan REJU




           School of Electrical and Electronic Engineering
                     Nanyang Technological University



10:15 AM                                                     1
                                Introduction
      Convolutive
      Blind Source Separation

                                           • Mixing process:
s1                                                                               
                                               x1 (t )   h11 (l ) s1 (t  l )   h12 (l ) s2 (t  l )
                                                         l 0                    l 0

                                                                                 
                                               x2 (t )   h21 (l ) s1 (t  l )   h22 (l ) s2 (t  l )
                                                         l 0                    l 0

s2

                                           • Unmixing process:
                                                         L                         L
                                            y1 (t )   w11 (l ) x1 (t  l )   w12 (l ) x2 (t  l )
                                                        l 0                     l 0

                                                         L                         L
                                            y2 (t )   w21 (l ) x1 (t  l )   w22 (l ) x2 (t  l )
                                                        l 0                      l 0




     10:15 AM                                                                                  2
                              Introduction
Convolutive Blind Source Separation    Instantaneous Blind Source Separation




 10:15 AM                                                              3
                                                 Introduction
Convolutive Blind Source Separation                           Instantaneous Blind Source Separation




     x1 (t )  h11 h12   s1 (t )                                  x1 (t )   h11 h12   s1 (t ) 
     x (t )  h         
                     h 22  s2 (t )                                  x (t )  h                  
     2   21                                                      2   21 h22  s2 (t )

    X(t )  H  S(t )                                                 X(t )  HS (t )
• In frequency domain:

  X 1 ( f , t )   H11 ( f ) H12 ( f )   S1 ( f , t ) 
  X ( f , t )   H ( f ) H ( f )  S ( f , t )
  2               21         22        2             

  X( f , t )  H ( f )S( f , t )
 10:15 AM                                                                                                   4
                Introduction
   s1                x1    No. of sources < No. of sensor
                     x2    Overdetermined mixing
   s2                x3

   s1                      No. of sources = No. of sensor
                     x1
                           Determined mixing
 s2                  x2
                     x3
  s3

      s1
                     x1    No. of sources > No. of sensor
  s2
                     x2    Underdetermined mixing
  s3       s4        x3
    s  4
10:15 AM                                                    5
      Approaches for BSS of Speech Signals

                Types of mixing



Instantaneous mixing      Convolutive mixing




10:15 AM                                       6
      Approaches for BSS of Speech Signals
                         Instantaneous mixing


  Step 1:   Selection of cost function


  Step 2:   Minimization or maximization of the cost function



                                         X1
                          S1                                   Y1
                                   H          W
                          S2                                   Y2
                                         X2

                                                  Separated?




10:15 AM                                                            7
         Approaches for BSS of Speech Signals
                             Instantaneous mixing
Selection of cost function
               Statistical independence                               Signals from two different sources are independent


                               Information theoretic                    Basic idea is      p y    pi  yi 
                                                                                                    i



                               Non-Gaussianity                          Central limit theorem:
                                                                        Mixture of two or more sources will be more
                                                                        Gaussian than their individual components
                                        Non Gaussianity measures:
                                           Kurtosis                                       
                                                                        kurt( y)  E y 4  3 E y 2
                                                                                                         2




                                           Negentropy                   J y   H y gauss   H y 
                                                                                  Entropy H y    py log py dy

                               Nonlinear cross moments                   E  f  yi   g  y j   0

               Temporal structure of speech Secondorder statsticscan be used
                                                eg., dioganaliz e the output correlation simultaneously for different time lags

               Non-stationarity of speech       Signals are divided into blocks and
                                                the correlatio n matrices are simultaneo usly diagonaliz ed
   10:15 AM                                                                                                         8
         Approaches for BSS of Speech Signals

                          Instantaneous mixing
Minimization or maximization of the cost function

              simple gradient method

              Natural gradient method e.g. Informax ICA algorithm




              Newton’s method           e.g. FastICA




   10:15 AM                                                         9
           Approaches for BSS of Speech Signals
                                         Convolutive Mixing



                   Time Domain:                                 Frequency Domain:
       P   L 1
yq   wqp (l ) x p (t  l )   q  1, , Q
      p 1 l  0                                          X( f , t )  H( f )S( f , t )

                                                          Y( f , t )  W( f ) X( f , t )
Advantage:                                            Advantage:
    No permutation problem                                 Low computational cost
Disadvantage:                                              Fast convergence
    Slow convergence                                  Disadvantage:
    High computational cost for long filter taps           Permutation Problem
                                                                       X1
                                                          S1                      Y1            Y2
                                                                   H      W                or
                                                          S2                      Y2            Y1
                                                                       X2

  10:15 AM                                                                                 10
                      Permutation Problem
                    in Frequency Domain BSS

                  One frequency bin                      Corresponding to y3
                              Instantaneous ICA algorithm


                            BSS
                                  f1




                                              permutation
x1                          BSS                                      y1               y1




                                                Problem
                                                Solving
              K point




                                                 K point
  x2                                                                 y2               y2




                                                  IFFT
                                  f2
                FFT




    x3                                                               y3               y3
                            BSS
Mixed                                                       Still signals      Separated
signals                           fk                        are mixed          signals



                    Corresponding to different sources
                    Due to permutation problem

   10:15 AM                                                                      11
                                           Motivation
                                         Instantaneous
                       Determined/
                      Overdetermined
                                                          Frequency      Frequency bin-    Permutation
                # mixtures ≥ # sources                     domain        wise separation     problem
                                          Convolutive

                                                         Time domain

  BSS
                                                         Mixing matrix       Source
                                         Instantaneous
                                                          estimation       estimation


                  Underdetermined

                # mixtures < # sources                    Frequency      Frequency bin-    Permutation
                                                           domain        wise separation     problem
                                          Convolutive
Automatic detection
  of no. of sources
                                                         Time domain




   10:15 AM                                                                                      12
                                   My Contribution - I
                                         Instantaneous
                       Determined/
                      Overdetermined
                                                          Frequency      Frequency bin-    Permutation
                # mixtures ≥ # sources                     domain        wise separation     problem
                                          Convolutive

                                                         Time domain

  BSS
                                                         Mixing matrix       Source
                                         Instantaneous
                                                          estimation       estimation


                  Underdetermined

                # mixtures < # sources                    Frequency      Frequency bin-    Permutation
                                                           domain        wise separation     problem
                                          Convolutive
Automatic detection
  of no. of sources
                                                         Time domain




   10:15 AM                                                                                      13
     Algorithm for Solving the Permutation
                    Problem

               One frequency bin
                           Instantaneous ICA algorithm


                        BSS
                              f1




                                         permutation
x1                      BSS                                                    y1




                                           Problem
                                           Solving
           K point




                                                            K point
  x2                                                                           y2




                                                             IFFT
                              f2
             FFT




    x3                                                                         y3
                        BSS
Mixed                                                                 Separated
signals                       fk                                        signals


                                        Permutation problem solved
                        Permutation problem


10:15 AM                                                                  14
                     Existing Method for
              Solving the Permutation Problem
 Direction Of Arrival (DOA) method:




Y1 ( f k )  W11 ( f k ) W12 ( f k )   X 1 ( f k ) 
Y ( f )  W ( f ) W ( f )  X ( f )
 2 k   21 k              22     k  2          k 


                                                                               Direction of y1 = -30o
                       2
                                                                               Direction of y2 = 20o
  U q  f k ,     Wqp  f k e
                                          j 2f k c 1d p sin( )

                      p 1
                                                              Position of the pth sensor
   10:15 AM                                                   Velocity of sound                         15
                   Existing Method for
            Solving the Permutation Problem
Direction Of Arrival (DOA) method:
                                     Disadvantages:
                                        Fails at lower frequencies.
                                        Fails when sources are near.
                                        Room reverberation.
                                        Sensor positions must be known.
                                     Reasons for failure at lower freq:
                                        Lower spacing causes error in
                                         phase difference measurement.
                                        The relation is approximated for
                                         plane wave front under
                                         anechoic condition



 10:15 AM                                                            16
                      Existing Method for
               Solving the Permutation Problem
Adjacent bands correlation method:
                                                          High
                 Low                                   correlation
              correlation
                                                          Low
                                                       correlation
                              BSS
                                    f1




                                         permutation
  x1                          BSS                                                 y1




                                           Problem
                                           Solving
                    K point




                                                              K point
    x2                                                                            y2




                                                               IFFT
                                    f2
                      FFT




      x3                                                                          y3
                              BSS
  Mixed                                                                 Separated
  signals                           fk                                  signals




   10:15 AM                                                                  17
                Existing Method for
         Solving the Permutation Problem
Adjacent bands correlation method:
                           r11       r11          r11     r11


    s1        ……..   K-1         K         K+1          K+2      K+3      ……..             Correlation matrix

                      r12        r12       r12          r12
                                                                                                  r11      r12
                      r21        r21       r21          r21
    s2        ……..   K-1         K         K+1          K+2      K+3      ……..
                                                                                                  r21       r22

                           r22       r22          r22     r22

     With confidence                 Example              Without confidence          Example
   r11 and r22  r12 and r21          0.1 0.8
                                     0.9 0.2
                                                              r11  r22  r12  r21   0.4 0.1
                                                                                      0.9 0.2         Change permutation
                                                                                           

   r11 and r22  r12 and r21         0.9 0.1                r11  r22  r12  r21   0.9 0.2           No change
                                     0.2 0.8                                        0.4 0.1
   10:15 AM                                                                                                   18
                Existing Method for
         Solving the Permutation Problem
Adjacent bands correlation method:

                           r11       r11     r11     r11
                                                                        Correlation matrix
         s1   ……..   K-1         K         K+1     K+2     K+3   ……..
                                                                          r11     r12
                     r12         r12       r12     r12
                     r21         r21       r21     r21                    r21      r22
         s2   ……..   K-1         K         K+1     K+2     K+3   ……..


                           r22       r22     r22     r22

Disadvantage: The method is not robust



   10:15 AM                                                                              19
                    Existing Method for
             Solving the Permutation Problem
Combination of DOA and Correlation methods method:




                DOA + Harmonic Correlation + Adjacent bands correlation

                Advantage: Increased robustness




  10:15 AM                                                                20
Proposed algorithm: Partial separation method
                                                           (Parallel configuration)
Reference: V. G. Reju, S. N. Koh and I. Y. Soon, “Partial separation method for solving permutation problem in frequency domain blind source separation of speech signals,” Neurocomputing,
                                                                         Vol. 71, NO. 10–12, June 2008, pp. 2098–2112.


                                           Time domain stage
                                                                                                                                                                                              y1
                                                                                                                                                                                              y2
                                                                                                                                                                                              x1

                                                                                                                                                                                              x2
                                                                                                                                                                                              ˆ
                                                                                                                                                                                              s1
                                                                                                                                                                                              ˆ
                                                                                                                                                                                              s2




10:15 AM                                                                                                                                                                                21
                                  Frequency domain stage
           Partial separation method
                          (Parallel configuration)
                 Time domain stage




10:15 AM                                             22
           Frequency domain stage
                 Partial separation method
                                    (Cascade configuration)
                                                              Parallel configuration


                        Frequency domain stage

    Time domain stage




10:15 AM                                                                       23
 Advantages of Partial Separation method

• Robustness




10:15 AM                               24
           Comparison with Adjacent Bands
                Correlation Method




10:15 AM                                    25
                  Comparison with DOA method




PS - Partial Separation method with confidence check, C1 - Correlation between the adjacent bins without confidence
check, C2 - Correlation between adjacent bins with confidence check, Ha - Correlation between the harmonic
components with confidence check, PS1 - Partial separation method alone without confidence check.
   10:15 AM                                                                                                 26
                                   My Contribution -II
                                         Instantaneous
                       Determined/
                      Overdetermined
                                                          Frequency      Frequency bin-    Permutation
                # mixtures ≥ # sources                     domain        wise separation     problem
                                          Convolutive

                                                         Time domain

  BSS
                                                         Mixing matrix       Source
                                         Instantaneous
                                                          estimation       estimation


                  Underdetermined

                # mixtures < # sources                    Frequency      Frequency bin-    Permutation
                                                           domain        wise separation     problem
                                          Convolutive
Automatic detection
  of no. of sources
                                                         Time domain




   10:15 AM                                                                                      27
  Underdetermined Blind Source Separation
        of Instantaneous Mixtures



                                                   S1 (k2 , t2 )  0, S2 (k2 , t2 )  0
S1 (k1 , t1 )  0, S 2 (k1 , t1 )  0
                             x1                   X 1 k , t 

                                                   S1 (k2 , t2 )  0, S2 (k2 , t2 )  0
S1 (k1 , t1 )  0, S 2 (k1 , t1 )  0

                           x2                   X 2 k , t 
                                        k
                                            t
  10:15 AM                                                                          28
                        Mathematical Representation of
                            Instantaneous Mixing
   Reference: V. G. Reju, S. N. Koh and I. Y. Soon, “An algorithm for mixing matrix estimation in instantaneous blind source separation,” Signal Processing, Vol. 89, Issue 9, September 2009, pp.
                                                                                             1762–1773.



Time domain:
                  x1 (t )   h11  h1Q   s1 (t ) 
                           
                                                                                                                                                    P – No. of mixtures
                  xP (t ) hP1  hPQ   sQ (t )
                                                
                                                                                                                                                          Q – No. of sources
Time-Frequency domain:
                  X 1 (k , t )   h11  h1Q   S1 (k , t ) 
                           
                                                         
                  X P (k , t ) hP1  hPQ   SQ (k , t )
                                                         
                                                                 h11                   h1q                    h1Q 
                                                                                                                
                                                                  S1 (k , t )       S q (k , t )       S Q (k , t )
                                                                 
                                                                hP1 
                                                                                      hPq                    hPQ 
                                                                                                                

                                        X(k , t )  h1S (k , t )    h 2 S (k , t )    h Q S (k , t )

  10:15 AM                                                                                                                                                                                     29
                                  Single Source Points in
                                 Time-Frequency domain




Case 1 : at point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0   Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0

 Single source point 1                                                   Single source point 2

X(k1 , t1 )  h1S1 (k1 , t1 )  h0 2 (k1 , t1 )
                                 2S                                    X(k2 , t2 )  h1S1 (k2 , t2 )  h 2 S 2 (k2 , t2 )
                                                                                            0
 RX(k1 , t1 )  h1RS1 (k1 , t1 )                                   RX(k2 , t2 )  h 2 RS2 (k2 , t2 )
 I 10:15kAMt1 )  h1I S1 (k1 , t1 )
    X( 1 ,                                                             I X(k2 , t2 )  h 2 I S2 (k2 , t2 )                      30
                                  Single Source Points in
                                 Time-Frequency domain

                    Q
 Xk , t    h q S q k , t 
                   q 1



 Let         Q2


Case 1 : at point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0   Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0

 Single source point 1                                                   Single source point 2

 X(k1 , t1 )  h1S1 (k1 , t1 )                                        X(k2 , t2 )  h 2 S 2 (k2 , t2 )

 RX(k1 , t1 )  h1RS1 (k1 , t1 )                                   RX(k2 , t2 )  h 2 RS2 (k2 , t2 )
 I 10:15kAMt1 )  h1I S1 (k1 , t1 )
    X( 1 ,                                                             I X(k2 , t2 )  h 2 I S2 (k2 , t2 )                      31
                                    Single Source Points in
                                   Time-Frequency domain

  Case 1 : at point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0   Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0

   Single source point 1                                                   Single source point 2

   X(k1 , t1 )  h1S1 (k1 , t1 )                                        X(k2 , t2 )  h 2 S 2 (k2 , t2 )

   RX(k1 , t1 )  h1RS1 (k1 , t1 )                                   RX(k2 , t2 )  h 2 RS2 (k2 , t2 )
   I X(k1 , t1 )  h1I S1 (k1 , t1 )                                 I X(k2 , t2 )  h 2 I S2 (k2 , t2 )

    .·. At single source point 1:                                              .·.   At single source point 2:

Direction of RX(k1 , t1 )  Direction of h1                             Direction of RX(k2 , t2 )  Direction of h 2
Direction of I X(k1 , t1 )  Direction of h1                            Direction of I X(k2 , t2 )  Direction of h 2
Direction of RX(k1 , t1  Direction of I X(k1 , t1                  Direction of RX(k2 , t2   Direction of I X(k2 , t2 
      10:15 AM                                                                                                                        32
          Scatter Diagram of the Mixtures When
                Source are Perfectly Sparse

 Example:




 X 1 (k ,1)
 X (k ,1)
               X 1 (k ,2)   X 1 (k ,3)   X 1 (k ,4)                                                                   0        0
                                                      X 1 (k ,5)   h11 h12   S1 (k ,1) S1 (k ,2) S1 (k ,3) S1 (k ,4) S1 (k ,5) 
                                                                                           0
 2            X 2 (k ,2)   X 2 (k ,3)   X 2 (k ,4)   X 2 (k ,5) h21 h22  S 2 (k ,1) S 2 (k ,2) S 2 (k ,3) S 2 (k ,4) S 2 (k ,5)
                                                                              0                     0                           


                                                                                   h 
                                                                              h1   11                     h 
                                                                                   h21               h 2   12 
                                                                                                             h22 




     10:15 AM                                                                                                                  33
          Scatter Diagram of the Mixtures When
             Source are Not Perfectly Sparse

 Example:




 X 1 (k ,1)
 X (k ,1)
               X 1 (k ,2)   X 1 (k ,3)   X 1 (k ,4)                                                                   0        0
                                                      X 1 (k ,5)   h11 h12   S1 (k ,1) S1 (k ,2) S1 (k ,3) S1 (k ,4) S1 (k ,5) 
                                                                                           0
 2            X 2 (k ,2)   X 2 (k ,3)   X 2 (k ,4)   X 2 (k ,5) h21 h22  S 2 (k ,1) S 2 (k ,2) S 2 (k ,3) S 2 (k ,4) S 2 (k ,5)
                                                                              0                     0                           


                                                                                   h 
                                                                              h1   11                     h 
                                                                                   h21               h 2   12 
                                                                                                             h22 




     10:15 AM                                                                                                                  34
     Scatter Diagram of the Mixtures when
               Sources are Sparse


                           h4        h3
No. of sources = 6              h1
                      h2
No. of mixtures = 2                       h6


                                          h5




10:15 AM                                       35
     Scatter Diagram of the Mixtures when
      Sources are Sparse, After Clustering


                           h4        h3
No. of sources = 6              h1
                      h2
No. of mixtures = 2                       h6


                                          h5




10:15 AM                                       36
        Scatter Diagram of the Mixtures when
           Sources are Not Perfectly Sparse
Objective:
Estimation of the single source points.

                                     h4        h3
 No. of sources = 6                       h1
                             h2
 No. of mixtures = 2                                h6


                                                    h5




  10:15 AM                                               37
          Principle of the Proposed Algorithm for
           the Detection of Single Source Points

Case 1 : At point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0 Case 2 : At point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0

  Single source point 1                                                 Single source point 2

  RX(k1 , t1 )  h1RS1 (k1 , t1 )                                  RX(k2 , t2 )  h 2 RS2 (k2 , t2 )
  I X(k1 , t1 )  h1I S1 (k1 , t1 )                                I X(k2 , t2 )  h 2 I S2 (k2 , t2 )
Case 3 : At point k3 , t3 , S1 k3 , t3   0 , S 2 k3 , t3   0
  Multi source point

    X(k3 , t3 )  h1S1 (k3 , t3 )  h 2 S 2 (k3 , t3 )
    RX(k3 , t3 )  h1RS1 (k3 , t3 ) h 2 RS2 (k3 , t3 )
    I X(k3 , t3 )  h1I S1 (k3 , t3 ) h 2 I S2 (k3 , t3 )
     10:15 AM                                                                                                                     38
          Principle of the Proposed Algorithm for
           the Detection of Single Source Points

Case 1 : At point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0 Case 2 : At point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0

  Single source point 1                                                 Single source point 2

  RX(k1 , t1 )  h1RS1 (k1 , t1 )                                  RX(k2 , t2 )  h 2 RS2 (k2 , t2 )
  I X(k1 , t1 )  h1I S1 (k1 , t1 )                                I X(k2 , t2 )  h 2 I S2 (k2 , t2 )
Case 3 : At point k3 , t3 , S1 k3 , t3   0 , S 2 k3 , t3   0
  Multi source point

    X(k3 , t3 )  h1S1Rk3Xt3 ) ,h 2) kDirection of I X (k , t )
    Direction of (  , (k3 t3 S 2 ( 3 , t3 )                   3 3
                       RS (k           RS (k , t )
    RX(k3 , t3 )  h1RS 1(k3 ,,tt3 )) h 2 RS2(k 3 , t3 )
     if and only if                               1      3     3
                                                                                   2       3     3
    I X(k3 , t3 )  h1I S1 ((k , t3 ) h 2 IIS2 ((k, t,3t) 
                        I S k3 , t )            S k3 )
                                                 1      3      3                    2      3      3
     10:15 AM                                                                                                                     39
    Principle of the Proposed Algorithm for
     the Detection of Single Source Points
                                                 Average of 15 pairs of speech utterances
                                                 of length 10 s each


                                Direction of RX (k3 , t3 )  Direction of I X (k3 , t3 )
                                                  RS1 (k3 , t3 ) RS 2 (k3 , t3 )
                                if and only if                     
                                                  I S1 (k3 , t3 ) I S 2 (k3 , t3 )




              Direction of RX (k , t )  Direction of I X (k , t )
              Direction of RX (k , t )  Direction of I X (k , t )




10:15 AM                                                                                    40
      Proposed Algorithm for the Detection of
               Single Source Points
                                X 1 (k1 , t1 )

                            x1                                       X 1 k , t 


                                X 2 (k1 , t1 )


                           x2                                      X 2 k , t              RX 1 (k1 , t1 )
                                          k                                                 RX (k , t )
                                                                                                2   1 1 

                                                       t                             I X 1 (k1 , t1 )
                                                                                     I X (k , t )
                                                                                          2   1 1 


                                                                                            RX 1 (k1 , t1 )
                                                                                            RX (k , t )
                  RX 1 (k1 , t1 ) jI X 1 (k1 , t1 )                                        1 1 
                                                             
                                                                                                 2
                                                         
                  RX 2 (k1 , t1 ) jI X 2 (k1 , t1 )                              I X 1 (k1 , t1 )
                                                                                          I X (k , t )
         RX (k , t ) I X (k , t )
                    T                                                                          2   1 1 
                                       cos 
        RX (k , t ) I X (k , t )
    10:15 AM                                                                                            41
           Elimination of Outliers


                  SSPs detection




                                     Clustering
                     Outlier
                   elimination




10:15 AM                             42
                      Experimental Results




                                         NMSE  47.67dB




 No. of mixtures =2, No. of sources =6


10:15 AM                                                   43
           Detected Single Source Points,
                  Three mixtures




                 No. of mixtures =3, No. of sources =6
10:15 AM                                                 44
 Comparison with Classical Algorithms for
           Determined Case

            Average of 500 experimental results




                                                  No. of mixtures =2
                                                  No. of sources =2




                                                                     ->
10:15 AM                                                        45
Comparison with Method Proposed in [1],
        Underdetermined case
           Normalized mean square error (NMSE)
              in mixing matrix estimation (dB)




                                                                                                P – No. of mixtures
                                                                                                Q – No. of sources
                                                 Order of the mixing matrices (PxQ)
 [1] Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, “Underdetermined blind source separation based on
 sparse representation,” IEEE Transactions on Signal Processing, vol. 54, p. 423–437, Feb. 2006.

10:15 AM                                                                                                  46
     Advantages of the Proposed algorithm

  1) Much simpler constrain: the algorithm does not require “single source zone”.
  2) Separation performance is better.
  3) The algorithm is extremely simple but effective

       Step 1:   Convert x in the time domain to the TF domain to get X.
       Step 2:   Check the condition
                   RX (k , t ) I X (k , t )
                              T
                                                 cos 
                  RX (k , t ) I X (k , t )

       Step 3:   If the condition is satisfied, then X(k, t) is a sample at
                 the SSP, and this sample is kept for mixing matrix estimation;
                 otherwise, discard the point.
       Step 4:    Repeat Steps 2 to 3 for all the points in the TF plane
                 or until sufficient number of SSPs are obtained.


                                                                                         ->
10:15 AM                                                                            47
              My Contributions – III, IV and V
                                         Instantaneous
                       Determined/
                      Overdetermined
                                                          Frequency      Frequency bin-    Permutation
                # mixtures ≥ # sources                     domain        wise separation     problem
                                          Convolutive

                                                         Time domain

  BSS
                                                         Mixing matrix       Source
                                         Instantaneous
                                                          estimation       estimation


                  Underdetermined

                # mixtures < # sources                    Frequency      Frequency bin-    Permutation
                                                           domain        wise separation     problem
                                          Convolutive
Automatic detection
  of no. of sources
                                                         Time domain




   10:15 AM                                                                                      48
        Underdetermined Convolutive Blind Source
         Separation via Time-Frequency Masking
    Reference: V. G. Reju, S. N. Koh and I. Y. Soon, “Underdetermined Convolutive Blind Source Separation via Time- Frequency Masking,” IEEE Transactions on Audio, Speech and Language Processing, Vol.
                                                                                       18, NO. 1, Jan. 2010, pp. 101–116.




                                                                                                                                                                                                  Y1 ( k , t )
Mic 1
             STFT




                                                                                                              Apply
                                                                                                              mask
                      k
                                                                                                                                                                                                  YQ ( k , t )
                                        t                             X 1 (k , t )

             Mixture in TF domain
                                                                       X P (k , t )                                                                                                               Y1 ( k , t )
              STFT




                                                                                                              Apply
                                                                                                              Mask
Mic P                 k                                                                                                                                                                           YQ ( k , t )

                                            t   

 PQ

                                     Mask estimation
        10:15 AM                                                                                                                    Separated signals in TF domain
                                                                                                                                                              49
                Mathematical Representation
Time domain:                 Q L 1
               x p (n)   hpq (l )sq (n  l )                           p  1,, P   P – No. of mixtures
                            q 1 l 0                                                  Q – No. of sources
Frequency domain:

                X 1 (k , t )   H11 (k )  H1Q (k )   S1 (k , t ) 
                                     
                                                      
                                                   
                                                                      
                              
                X P (k , t )  H P1 (k )  H PQ (k )  SQ (k , t )
                                                                 

                               H11 (k )              H1q (k )               H1Q (k ) 
                                                                                       
                                S1 (k , t )       S q (k , t )       S Q (k , t )
                                        
                               H P1 (k )
                                                     H Pq (k )              H PQ (k )
                                                                                       


                X(k , t )  H1 (k ) S1 (k , t )    H q (k ) S q (k , t )    H Q (k ) S Q (k , t )



  10:15 AM                                                                                          50
                                        Single source points
                                                   Instantaneous mixing
Case 1 : at point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0     Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
 Single source point 1                                                 Single source point 2

RX(k1 , t1 )  H1 (k ) RS1 (k1 , t1 )                             RX(k2 , t2 )  H 2 (k ) RS 2 (k2 , t2 )
I X(k1 , t1 )  H1 (k ) I S1 (k1 , t1 )                           I X(k2 , t2 )  H 2 (k ) I S2 (k2 , t2 )

                                                     Convolutive mixing
Case 1 : at point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0     Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
 Single source point 1                                                 Single source point 2


      X(k1 , t1 )  H1 (k ) S1 (k1 , t1 )                                 X(k2 , t2 )  H 2 (k ) S 2 (k2 , t2 )



    10:15 AM                                                                                                                          51
           Basic Principle of Single Source Points
                          Detection
                                                      Convolutive mixing
Case 1 : at point k1 , t1 , S1 k1 , t1   0 , S2 k1 , t1   0   Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
 Single source point 1                                                Single source point 2


      X(k1 , t1 )  H1 (k ) S1 (k1 , t1 )                               X(k2 , t2 )  H 2 (k ) S 2 (k2 , t2 )

                  H
    cos C  
                 U1 U 2
                U1 U 2                              U  UH U

                e j
                                                                                                                                    ->
      cos H   cosC                       0   H   2 and      
    H is called Hermitian angle
                                               The Hermitian angle between the complex
    is called pseudo angle
                                               vectors u1 and u2 will remain the same even if
                                               the vectors are multiplied by any complex
    10:15 AM                                   scalars, whereas the pseudo angle will change.
                                                                                         52
               Algorithm for Single Source Points
                           Detection
                       X 1 (k1 , t1 )
                                                                                     X 1 k , t 
                   x1
                                 k                                                       k1
                                                                                               θH2
                                                                                               θH1
                       X 2 (k1 , t1 )

                                                                                    X 2 k , t 
                  x2                                                                                             1  j1
                                                                                                               r      
                                 k                                                       k1                      1  j1

                                        t1            t                                        θH1     X 1 (k1 , t1 ) 
                                                                                                       X (k , t )
                             H (k )                                                                  2 1 1 
                            11 1  S1 (k1 , t1 ),             
                             H 21 (k1 )
                                                                                                         1  j1
                                                                                                     r      
                                                             X(k1 , t1 ) H r                           1  j1
                       OR                  H        cos 
                                                           1                   
                                                q          X(k1 , t1 ) r     
                                                                               
                            H (k )                                                     θH2          X 1 (k1 , t1 ) 
                           12 1  S 2 (k1 , t1 ),                                                  X (k , t )
                            H 22 (k1 )                                                              2 1 1 
    10:15 AM                                                                                                               53
           Mask Estimation by k-means (KM)




                                                                             Clean

                Yq (k , t )  M q (k , t ) X p (k , t )   t , q  1, , Q

                                                                         Estimated



10:15 AM                                                                             54
 Mask Estimation by Fuzzy c-means (FCM)




                                                                        Clean

           Yq (k , t )  M q (k , t ) X p (k , t )   t , q  1, , Q

                                                                   Estimated



10:15 AM                                                                        55
Automatic Detection of Number of Sources




    Cluster validation technique:
     For c = 2 to cmax
               Cluster the data into c clusters.
               Calculate the cluster validation index.
     End
     Take c corresponding to the best cluster as the
     number of sources.



                                                          ->
10:15 AM                                                 56
           Elimination of Low Energy Points




10:15 AM                                      57

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:34
posted:8/21/2012
language:English
pages:57