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 As
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
uCn 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.