Transcript slides

Audio Demixing with Decorrelation, Cross
Cancellation, Normalization, and
Regularization
Sean Webster
Mentors: Ernie Esser, Jack Xin
The Problem
x1  a11  a12 s1 
  
 
x 2  a21  a22 s2 
x1  a11  s1  a12  s2
x 2  a21  s1  a22  s2

Partial Inversion
x  As
1  a22  a12 
A 


det A a21  a11  
1
A 1 x  s
 a22  a12 x1 
s1  v1 

  det A   
a21  a11  x 2 
s2  v 2 
v1  a22  x1  a12  x 2
v 2  a21  x1  a11  x 2

Decorrelation
E[s1(t)s2 (t  n)]  0
E[v1(t)v 2 (t  n)]  0
a22 a21 E[x1 (t)x1 (t  n)] a12 a21 E[x 2 (t)x1 (t  n)]
a22 a11 E[x1 (t)x 2 (t  n)]  a12 a11 E[x 2 (t)x 2 (t  n)]  0
Cnij  E[x i (t)x j (t  n)]
21
12
22
a22 a21C11

a
a
C

a
a
C

a
a
C
n
12 21 n
22 11 n
12 11 n  0
12 
C11
a12 
C
n
n
a22 a21 C 21 C 22 a  0
 11 
 n
n 
uCn w  0


l1 Normalization Constraint
 1  0
 w 1  0
 u
2
2
2
l1
2
2
2
l1
F   u Cn w  

n
2
2
u
2
l1

2
1  
2
w
2
l1

2
1  0
Cross Cancellation
x11  a11  a12 s1 
  
 
x12  a21  a22 0 
x 21  a11  a12 0 
  
 
x 22  a21  a22 s2 
x11  a11  s1
x12  a21  s1
x 21  a12  s2
x 22  a22  s2
a21  x11  a21  a11  s1
a11  x12  a11  a21  s1
a22  x 21  a22  a12  s2
a12  x 22  a12  a22  s2
a21  x11  a11  x12  0
a22  x 21  a12  x 22  0
Regularization
r  exp(c *[0 : q 1])
r r 
R  

r r 
p  aA.* R

Results
Instantaneous A
Cross Cancellation A
Results
Cross Cancellation + Normalization +
Regularization A
Cross Cancellation + Normalization +
Regularization + Decorrelation A
Results
Convoluted A
Cross Cancellation A
Results
Cross Cancellation + Normalization
A
References
Alexis Favrot, Christof Faller, and Fabian Kuech. Reverberation modeling in acoustic
echo suppression. IEEE Workshop on Applications of Signal Processing to Audio and
Acoustics, 2011.
Jie Liu, Jack Xin, Yingyong Qi, and Fan-Gang Zheng. A time domain algorithm for blind
separation of convolutive sound mixtures and L1 constrainted minimization of
cross correlations. Communications in Mathematical Sciences, 7(1):109–128, 2009.
Meng Yu, Wenye Ma, Jack Xin, and Stanley Osher. A convex speech extraction model
and fast compu- tation by the split bregman method. Pages 1–8, 2010.