Blind Separation of Speech Mixtures
Vaninirappuputhenpurayil Gopalan REJU
School of Electrical and Electronic Engineering
Nanyang Technological University
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1
Introduction
Blind
Convolutive
Source Separation
• Mixing process:
s1


l 0
l 0


l 0
l 0
x1 (t )   h11 (l ) s1 (t  l )   h12 (l ) s2 (t  l )
x2 (t )   h21 (l ) s1 (t  l )   h22 (l ) s2 (t  l )
s2
• Unmixing process:
L
L
l 0
l 0
L
L
l 0
l 0
y1 (t )   w11 (l ) x1 (t  l )   w12 (l ) x2 (t  l )
y2 (t )   w21 (l ) x1 (t  l )   w22 (l ) x2 (t  l )
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2
Introduction
Convolutive Blind Source Separation
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Instantaneous Blind Source Separation
3
Introduction
Convolutive Blind Source Separation
Instantaneous Blind Source Separation
 x1 (t )  h11 h12   s1 (t ) 
 x (t )  h
  s (t )
h
22   2
 2   21

 x1 (t )   h11 h12   s1 (t ) 
 x (t )  h


 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 )
22
 2
  21
 2

X( f , t )  H ( f )S( f , t )
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4
Introduction
s1
x1
x2
x3
s2
s1
x1
x2
x3
s2
s3
No. of sources < No. of sensor
Overdetermined mixing
No. of sources = No. of sensor
Determined mixing
s1
s2
s3
s
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s4
x1
x2
x3
No. of sources > No. of sensor
Underdetermined mixing
5
Approaches for BSS of Speech Signals
Types of mixing
Instantaneous mixing
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Convolutive mixing
6
Approaches for BSS of Speech Signals
Instantaneous mixing
Step 1:
Selection of cost function
Step 2:
Minimization or maximization of the cost function
S1
S2
X1
H
Y1
Y2
W
X2
Separated?
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7
Approaches for BSS of Speech Signals
Instantaneous mixing
Selection of cost function
Signals from two different sources are independent
Statistical independence
py    pi  yi 
Information theoretic
Basic idea is
Non-Gaussianity
Central limit theorem:
Mixture of two or more sources will be more
Gaussian than their individual components
Non Gaussianity measures:
i
    
Kurtosis
kurt( y)  E y 4  3 E y 2
Negentropy
J y   H y gauss   H y 
2
Entropy H y     py  log py dy
Nonlinear cross moments
Ef  yi   g  y j   0
Temporal structure of speech Second order statstics can be used
eg., dioganaliz e the output correlatio n simultaneo usly for different time lags
Non-stationarity of speech
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Signals are divided into blocks and
the correlatio n matrices are simultaneo usly diagonaliz ed
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
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e.g. FastICA
9
Approaches for BSS of Speech Signals
Convolutive Mixing
P L 1
Time Domain:
yq   wqp (l ) x p (t  l )
q  1,, Q
p 1 l 0
Frequency Domain:
X( f , t )  H( f )S( f , t )
Y ( f , t )  W ( f ) X( f , t )
Advantage:
No permutation problem
Disadvantage:
Slow convergence
High computational cost for long filter taps
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Advantage:
Low computational cost
Fast convergence
Disadvantage:
Permutation Problem
X1
S1
Y1
H
W
S2
Y2
X2
or
10
Y2
Y1
Permutation Problem
in Frequency Domain BSS
Corresponding to y3
One frequency bin
Instantaneous ICA algorithm
x1
x2
x3
Mixed
signals
K point
FFT
f1
BSS
f2
BSS
fk
Solving
K point
permutation
IFFT
Problem
BSS
y1
y2
y3
y1
y2
y3
Still signals
are mixed
Separated
signals
Corresponding to different sources
Due to permutation problem
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11
Motivation
Instantaneous
Determined/
Overdetermined
Frequency
domain
# mixtures ≥ # sources
Frequency binwise separation
Permutation
problem
Convolutive
Time domain
BSS
Instantaneous
Mixing matrix
estimation
Source
estimation
Frequency
domain
Frequency binwise separation
Underdetermined
# mixtures < # sources
Permutation
problem
Convolutive
Automatic detection
of no. of sources
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Time domain
12
My Contribution - I
Instantaneous
Determined/
Overdetermined
Frequency
domain
# mixtures ≥ # sources
Frequency binwise separation
Permutation
problem
Convolutive
Time domain
BSS
Instantaneous
Mixing matrix
estimation
Source
estimation
Frequency
domain
Frequency binwise separation
Underdetermined
# mixtures < # sources
Permutation
problem
Convolutive
Automatic detection
of no. of sources
6:09 PM
Time domain
13
Algorithm for Solving the Permutation
Problem
One frequency bin
Instantaneous ICA algorithm
Mixed
signals
BSS
f2
BSS
fk
y1
y2
y3
K point
IFFT
x1
x2
x3
K point
FFT
f1
Solving
permutation
Problem
BSS
Separated
signals
Permutation problem solved
Permutation problem
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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 )
22
k  2
k 
 2 k   21 k
2
U q  f k ,   Wqp  f k e
p 1
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j 2f k c 1d p sin( )
Direction of y1 = -30o
Direction of y2 = 20o
Position of the pth sensor
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:
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
Lower spacing causes error in
phase difference measurement.

The relation is approximated for
plane wave front under
anechoic condition
16
Existing Method for
Solving the Permutation Problem
Adjacent bands correlation method:
High
correlation
Low
correlation
Low
correlation
Mixed
signals
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BSS
f2
BSS
fk
y1
y2
y3
K point
IFFT
x1
x2
x3
K point
FFT
f1
Solving
permutation
Problem
BSS
Separated
signals
17
Existing Method for
Solving the Permutation Problem
Adjacent bands correlation method:
r11
r11
s1
s2
……..
……..
r11
K-1
K
r12
r12
r12
r12
r21
r21
r21
r21
K-1
K
r22
With confidence
K+1
K+1
r22
Example
r11 and r22  r12 and r21
 0.1 0.8
0.9 0.2


r11 and r22  r12 and r21
0.9 0.1
0.2 0.8


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r11
r22
K+2
K+3
K+2
K+3
……..
Correlation matrix
……..
r11
r12
r21
r22
r22
Without confidence
r11  r22  r12  r21
r11  r22  r12  r21
Example
0.4 0.1
0.9 0.2


0.9 0.2
0.4 0.1


Change permutation
No change
18
Existing Method for
Solving the Permutation Problem
Adjacent bands correlation method:
r11
r11
r11
r11
Correlation matrix
s1
s2
……..
……..
K-1
K
r12
r12
r12
r12
r21
r21
r21
r21
K-1
K
r22
K+1
K+1
r22
r22
K+2
K+2
K+3
K+3
……..
r11
r12
r21
r22
……..
r22
Disadvantage: The method is not robust
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19
Existing Method for
Solving the Permutation Problem
Combination of DOA and Correlation methods method:
DOA + Harmonic Correlation + Adjacent bands correlation
Advantage: Increased robustness
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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
ŝ1
ŝ2
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Frequency domain stage
21
Partial separation method
(Parallel configuration)
Time domain stage
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Frequency domain stage
22
Partial separation method
(Cascade configuration)
Parallel configuration
Frequency domain stage
Time domain stage
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23
Advantages of Partial Separation method
• Robustness
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24
Comparison with Adjacent Bands
Correlation Method
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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.
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26
My Contribution -II
Instantaneous
Determined/
Overdetermined
Frequency
domain
# mixtures ≥ # sources
Frequency binwise separation
Permutation
problem
Convolutive
Time domain
BSS
Instantaneous
Mixing matrix
estimation
Source
estimation
Frequency
domain
Frequency binwise separation
Underdetermined
# mixtures < # sources
Permutation
problem
Convolutive
Automatic detection
of no. of sources
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Time domain
27
Underdetermined Blind Source Separation
of Instantaneous Mixtures
Mixture in
time
domain
Time to TF
domain
Detection
of SSPs
Mixing
matrix
estimation
Estimation
of Sources
S1 (k2 , t2 )  0, S2 (k2 , t2 )  0
S1 (k1 , t1 )  0, S2 (k1 , t1 )  0
X1 k , t 
x1
S1 (k2 , t2 )  0, S2 (k2 , t2 )  0
S1 (k1 , t1 )  0, S2 (k1 , t1 )  0
x2
X 2 k , t 
k
t
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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 ) 
          



 
 xP (t ) hP1  hPQ   sQ (t )
Time-Frequency domain:
 X 1 (k , t )   h11  h1Q   S1 (k , t ) 
          



 
 X P (k , t ) hP1  hPQ   SQ (k , t )
P – No. of mixtures
Q – No. of sources
 h1q 
 h1Q 
 h11 
 
 
    S1 (k , t )       S q (k , t )       SQ (k , t )
hPq 
hPQ 
hP1 
 
 
X(k , t )  h1S (k , t )    h 2 S (k , t )    h Q S (k , t )
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Single Source Points in
Time-Frequency domain
Case 1: at point k1, t1 , S1 k1, t1   0 , S2 k1, t1   0
Single source point 1
Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
Single source point 2
X(k1, t1 )  h1S1 (k1 , t1 )  h0
2 S2 (k1 , t1 )
X(k2 , t2 )  h1S1 (k20, t2 )  h2 S2 (k2 , t2 )
RX(k1 , t1 )  h1 RS1 (k1 , t1 )
RX(k2 , t2 )  h 2 RS 2 (k2 , t 2 )
I 6:09
X(PM
k1 , t1 )  h1 I S1 (k1 , t1 )
I X(k2 , t2 )  h 2 I S 2 (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
Single source point 1
Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
Single source point 2
X(k1, t1 )  h1S1 (k1 , t1 )
X(k2 , t2 )  h2 S2 (k2 , t2 )
RX(k1 , t1 )  h1 RS1 (k1 , t1 )
RX(k2 , t2 )  h 2 RS 2 (k2 , t 2 )
I 6:09
X(PM
k1 , t1 )  h1 I S1 (k1 , t1 )
I X(k2 , t2 )  h 2 I S 2 (k2 , t2 )
31
Single Source Points in
Time-Frequency domain
Case 1: at point k1, t1 , S1 k1, t1   0 , S2 k1, t1   0
Single source point 1
Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
Single source point 2
X(k1, t1 )  h1S1 (k1 , t1 )
X(k2 , t2 )  h2 S2 (k2 , t2 )
RX(k1 , t1 )  h1 RS1 (k1 , t1 )
RX(k2 , t2 )  h 2 RS 2 (k2 , t 2 )
I X(k1 , t1 )  h1 I S1 (k1 , t1 )
.·.
At single source point 1:
I X(k2 , t2 )  h 2 I S 2 (k2 , t2 )
.·.
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 RX(k1 , t1  Direction of I X(k1, t1
Direction of RX(k2 , t2   Direction of I X(k2 , t2 
Direction of I X(k1 , t1 )  Direction of h1
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Direction of I X(k2 , t 2 )  Direction of h 2
32
Scatter Diagram of the Mixtures When
Source are Perfectly Sparse
Example:
 X 1 (k ,1)
 X (k ,1)
 2
X 1 (k ,2)
X 1 (k ,3)
X 1 (k ,4)
X 2 (k ,2)
X 2 (k ,3)
X 2 (k ,4)
0
h 
h1   11 
h21 
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0
0
0
X 1 (k ,5)   h11 h12   S1 (k ,1) S1 (k ,2) S1 (k ,3) S1 (k ,4) S1 (k ,5) 

X 2 (k ,5) h21 h22  S 2 (k ,1) S2 (k ,2) S2 (k ,3) S2 (k ,4) S2 (k ,5)
0
h 
h 2   12 
h22 
33
Scatter Diagram of the Mixtures When
Source are Not Perfectly Sparse
Example:
 X 1 (k ,1)
 X (k ,1)
 2
X 1 (k ,2)
X 1 (k ,3)
X 1 (k ,4)
X 2 (k ,2)
X 2 (k ,3)
X 2 (k ,4)
0
h 
h1   11 
h21 
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0
0
0
X 1 (k ,5)   h11 h12   S1 (k ,1) S1 (k ,2) S1 (k ,3) S1 (k ,4) S1 (k ,5) 

X 2 (k ,5) h21 h22  S 2 (k ,1) S2 (k ,2) S2 (k ,3) S2 (k ,4) S2 (k ,5)
0
h 
h 2   12 
h22 
34
Scatter Diagram of the Mixtures when
Sources are Sparse
No. of sources = 6
No. of mixtures = 2
h4
h2
h1
h3
h6
h5
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35
Scatter Diagram of the Mixtures when
Sources are Sparse, After Clustering
No. of sources = 6
No. of mixtures = 2
h4
h2
h1
h3
h6
h5
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36
Scatter Diagram of the Mixtures when
Sources are Not Perfectly Sparse
Objective:
Estimation of the single source points.
No. of sources = 6
No. of mixtures = 2
h4
h2
h1
h3
h6
h5
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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
RX(k1 , t1 )  h1 RS1 (k1 , t1 )
I X(k1 , t1 )  h1 I S1 (k1 , t1 )
Single source point 2
RX(k2 , t2 )  h 2 RS 2 (k2 , t 2 )
I X(k2 , t2 )  h 2 I S 2 (k2 , t2 )
Case 3 : At point k3 , t3 , S1 k3 , t3   0 , S2 k3 , t3   0
Multi source point
X(k3 , t3 )  h1S1 (k3 , t3 )  h 2 S2 (k3 , t3 )
RX(k3 , t3 )  h1RS1 (k3 , t3 ) h 2 RS 2 (k3 , t3 )
I X(k3 , t3 )  h1I S1 (k3 , t3 ) h 2 I S 2 (k3 , t3 )
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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
RX(k1 , t1 )  h1 RS1 (k1 , t1 )
I X(k1 , t1 )  h1 I S1 (k1 , t1 )
Single source point 2
RX(k2 , t2 )  h 2 RS 2 (k2 , t 2 )
I X(k2 , t2 )  h 2 I S 2 (k2 , t2 )
Case 3 : At point k3 , t3 , S1 k3 , t3   0 , S2 k3 , t3   0
Multi source point
X
(k3 , t3 )  hof
(k3X, t3()k,ht 2)S2(kDirection
1S1R
3 , t3 )
Direction
of I X (k3 , t3 )
3 3
RX(k3 , t3 )  h1RRSS11((kk33,,tt33)) h 2RRSS22((kk33 ,, tt33))
if and only if

I X(k3 , t3 )  h1IISS11((kk33, t,3t)3)
 h 2 IISS22(k(k3 ,3t,3t)3)
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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 )
if and only if
RS1 (k3 , t3 ) RS 2 (k3 , t3 )

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 )
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40
Proposed Algorithm for the Detection of
Single Source Points
X1 (k1 , t1 )
X1 k , t 
x1
X 2 (k1 , t1 )
x2
X 2 k , t 
k
t






 RX 1 (k1 , t1 ) jI X 1 (k1 , t1 )
 

 RX 2 (k1 , t1 ) jI X 2 (k1 , t1 )
RX (k , t ) I X (k , t )
 cos 
RX (k , t ) I X (k , t )
T
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 RX 1 (k1 , t1 )
 RX (k , t )
2
1 1 

 I X 1 (k1 , t1 )
 I X (k , t )
2
1 1 


 RX 1 (k1 , t1 )
 RX (k , t )
2
1 1 

 I X 1 (k1 , t1 )
 I X (k , t )
2
1 1 

41
Elimination of Outliers
Clustering
SSPs detection
Outlier
elimination
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42
Experimental Results
NMSE  47.67dB
No. of mixtures =2, No. of sources =6
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43
Detected Single Source Points,
Three mixtures
No. of mixtures =3, No. of sources =6
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44
Comparison with Classical Algorithms for
Determined Case
Average of 500 experimental results
No. of mixtures =2
No. of sources =2
->
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45
Normalized mean square error (NMSE)
in mixing matrix estimation (dB)
Comparison with Method Proposed in [1],
Underdetermined case
Order of the mixing matrices (PxQ)
P – No. of mixtures
Q – No. of sources
[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.
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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:
Step 2:
Convert x in the time domain to the TF domain to get X.
Check the condition
RX (k , t ) I X (k , t )
 cos 
RX (k , t ) I X (k , t )
T
Step 3:
Step 4:
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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.
Repeat Steps 2 to 3 for all the points in the TF plane
or until sufficient number of SSPs are obtained.
47
->
My Contributions – III, IV and V
Instantaneous
Determined/
Overdetermined
Frequency
domain
# mixtures ≥ # sources
Frequency binwise separation
Permutation
problem
Convolutive
Time domain
BSS
Instantaneous
Mixing matrix
estimation
Source
estimation
Frequency
domain
Frequency binwise separation
Underdetermined
# mixtures < # sources
Permutation
problem
Convolutive
Automatic detection
of no. of sources
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Time domain
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 )
Apply
mask
STFT
Mic 1
k
YQ ( k , t )

t
X 1 (k , t )
Mixture in TF domain
Mic P
Y1 (k , t )
Apply
Mask
STFT
X P (k , t )
k
YQ ( k , t )
t

PQ
Mask estimation
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49
Separated signals in TF domain
Mathematical Representation
Time domain:
Q L 1
x p (n)   hpq (l ) sq (n  l )
q 1 l  0
p  1, , P
P – No. of mixtures
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 )
 H1q (k ) 
 H1Q (k ) 
 H11 (k ) 




    S1 (k , t )       S q (k , t )       SQ (k , t )
 H Pq (k )
 H PQ (k )
 H P1 (k )




X(k , t )  H1 (k ) S1 (k , t )    H q (k ) S q (k , t )    H Q (k ) SQ (k , t )
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Single source points
Instantaneous mixing
Case 1: at point k1, t1 , S1 k1, t1   0 , S2 k1, t1   0
Single source point 1
Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
Single source point 2
RX(k1 , t1 )  H1 (k ) RS1 (k1 , t1 )
I X(k1 , t1 )  H1 (k ) I S1 (k1 , t1 )
RX(k 2 , t 2 )  H 2 (k ) RS 2 (k 2 , t 2 )
I X(k 2 , t 2 )  H 2 (k ) I S 2 (k 2 , t 2 )
Convolutive mixing
Case 1: at point k1, t1 , S1 k1, t1   0 , S2 k1, t1   0
Single source point 1
X(k1 , t1 )  H1 (k )S1 (k1 , t1 )
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Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
Single source point 2
X(k2 , t2 )  H2 (k )S2 (k2 , t2 )
51
Basic Principle of Single Source Points
Detection
Convolutive mixing
Case 1: at point k1, t1 , S1 k1, t1   0 , S2 k1, t1   0
Single source point 1
Single source point 2
X(k1 , t1 )  H1 (k )S1 (k1 , t1 )
U1H U 2
cos C  
U1 U 2
Case 2 : at point k2 , t2 , S1 k2 , t2   0 , S2 k2 , t2   0
X(k2 , t2 )  H2 (k )S2 (k2 , t2 )
U  UH U
 e j
  cos H   cosC 
 H is called Hermitian angle
 is called pseudo angle
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0   H   2 and      
->
The Hermitian angle between the complex
vectors u1 and u2 will remain the same even if
the vectors are multiplied by any complex
52
scalars, whereas the pseudo angle will change.
Algorithm for Single Source Points
Detection
X1 (k1 , t1 )
x1
X1 k , t 
k
k1
θH1
X 2 (k1 , t1 )
X 2 k , t 
x2
k
k1
t
t1
θH1
 H (k ) 
  11 1  S1 (k1 , t1 ),
 H 21 (k1 )






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θH2
OR
H

 X(k1 , t1 ) H r
 cos 
 X(k1 , t1 ) r

1
q
 H (k ) 
  12 1  S 2 (k1 , t1 ),
 H 22 (k1 )

1  j1
r

1  j1
 X 1 (k1 , t1 ) 
 X (k , t )
 2 1 1 
1  j1
r

1  j1




θH2
 X 1 (k1 , t1 ) 
 X (k , t )
 2 1 1 
53
Mask Estimation by k-means (KM)
Clean
Yq (k , t )  M q (k , t ) X p (k , t )
t , q  1, , Q
Estimated
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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
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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.
->
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56
Elimination of Low Energy Points
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57