Audio Demixing with Decorrelation, Cross Cancellation, Normalization, and

Regularization

Sean WebsterMentors: Ernie Esser, Jack Xin

The Problem

€

x1

x2

⎛

⎝ ⎜

⎞

⎠ ⎟=a11 ∗ a12 ∗

a21∗ a22 ∗

⎛

⎝ ⎜

⎞

⎠ ⎟s1s2

⎛

⎝ ⎜

⎞

⎠ ⎟

x1 = a11 ∗s1 + a12 ∗s2x2 = a21∗s1 + a22 ∗s2

Partial Inversion

€

x = A∗s

A−1 =1

det A

a22 ∗ −a12 ∗

−a21∗ a11∗

⎛

⎝ ⎜

⎞

⎠ ⎟

A−1x = s

a22 ∗ −a12 ∗

−a21∗ a11∗

⎛

⎝ ⎜

⎞

⎠ ⎟x1

x2

⎛

⎝ ⎜

⎞

⎠ ⎟= det A

s1s2

⎛

⎝ ⎜

⎞

⎠ ⎟=v1

v2

⎛

⎝ ⎜

⎞

⎠ ⎟

v1 = a22 ∗x1 − a12 ∗x2

v2 = −a21∗x1 + a11∗x2

Decorrelation

€

E[s1(t)s2(t − n)] = 0

E[v1(t)v2(t − n)] = 0

−a22a21E[x1(t)x1(t − n)]+ a12a21E[x2(t)x1(t − n)]

+a22a11E[x1(t)x2(t − n)] − a12a11E[x2(t)x2(t − n)] = 0

Cnij = E[x i(t)x j (t − n)]

−a22a21Cn11 + a12a21Cn

21 + a22a11Cn12 − a12a11Cn

22 = 0

a22 a21( )−Cn

11 Cn12

Cn21 −Cn

22

⎛

⎝ ⎜

⎞

⎠ ⎟a12

a11

⎛

⎝ ⎜

⎞

⎠ ⎟= 0

uΤCnw = 0

l1 Normalization Constraint

€

σ2 ul1

2−1( )

2

= 0

σ 2 w l1

2−1( )

2

= 0

F = uΤCnw2

n

∑ +σ 2 ul1

2−1( )

2

+σ 2 wl1

2−1( )

2

= 0

Cross Cancellation

€

x11

x12

⎛

⎝ ⎜

⎞

⎠ ⎟=a11 ∗ a12 ∗

a21∗ a22 ∗

⎛

⎝ ⎜

⎞

⎠ ⎟s10

⎛

⎝ ⎜

⎞

⎠ ⎟

x21

x22

⎛

⎝ ⎜

⎞

⎠ ⎟=a11 ∗ a12 ∗

a21∗ a22 ∗

⎛

⎝ ⎜

⎞

⎠ ⎟0

s2

⎛

⎝ ⎜

⎞

⎠ ⎟

x11 = a11 ∗s1x12 = a21∗s1x21 = a12 ∗s2x22 = a22 ∗s2

a21∗x11 = a21∗a11 ∗s1a11 ∗x12 = a11 ∗a21∗s1a22 ∗x21 = a22 ∗a12 ∗s2a12 ∗x22 = a12 ∗a22 ∗s2

a21∗x11 − a11 ∗x12 = 0

a22 ∗x21 − a12 ∗x22 = 0

Regularization

€

r = exp(c *[0 :q −1])

R =r r

r r

⎛

⎝ ⎜

⎞

⎠ ⎟

p = aA.*R

Results

Instantaneous A Cross Cancellation A

ResultsCross Cancellation + Normalization + Regularization A

Cross Cancellation + Normalization + Regularization + Decorrelation A

Results

Convoluted A Cross Cancellation A

Results

Cross Cancellation + Normalization A

ReferencesAlexis 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.