Speech Separation Using Ica

SPEECH SEPARATION USING ICA



Correlation and Covariance matrix(with properties for dependent and independent vectors)

 Kurtosis  Hotelling

Transformation

 Orthonormality



First moment of a random vector.

 Correlation

vector.

between components of the



Covariance matrix is corresponding to correlation matrix

 Covariance

 Cross

matrix for two vectors x and y

correlation matrix



Measures the Gaussian nature of the signal

 Independent

signals are more Gaussian.

 Formula

is given as: (for column X1)

 Maximum

for independent vectors



Y1(t) and Y2(t) are dependent on each other

 Covariance

matrix is not a diagonal matrix for mixed signal vectors

 Using

Hotelling transformation Y1(t) Z1(t) , Y2(t)

 Column

Z2(t)

vectors of the transformation matrix is orthogonol in nature

Now, the covariance of LHS whereas the covariance of RHS

To get both sides equal, we require the columns of B orthonormal

Covariance of RHS

Covariance of LHS

CONSTRAINT 1 : Kurtosis values are maximum for independent signals. Thus, whatever equations we get for the coefficients of the independent signals, can be differentiated and principle of maxima can be applied. Thus, Lagrange’s method is used to estimate the values of B.



Using Hotelling transformation Y1(t) Z1(t) , Y2(t)

Z2(t)



Initializing matrix B such that BTB = [I]



Updating matrix B using Iterative formula b11 (t+1) = E[(b11 (t) z1i + b21 (t) z2i )3 z1i ]-3*b11 (t)



Making columns of matrix B orthonormal to each other B = B * real(inv(B'*B)^(1/2))



Repeating the above two steps ‘N’ times



Obtaining independent matrix signals by [BT ] * [Z] = [X]

 Centering

Finding central moments

 Whitening

Making coefficients uncorrelated



Denoising.

 Telecommunication(CDMA).  Reducing

noise in natural images(image processing)

 Finding

hidden factors in financial data(econometrics)



A. Hyvärinen, J. Karhunen, E. Oja (2001): Independent Component Analysis, New York: Wiley, ISBN 978-0471-40540-5



Case Studies of Independent Component Analysis – Numerical Analysis of Linear Algebra - Alan Oursland, Judah De Paula, Nasim Mahmood



Pierre Comon (1994): Independent Component Analysis: a new concept?, Signal Processing, Elsevier, 36(3):287--314 (The original paper describing the concept of ICA)



FastICA toolbox version 2.5 - www.cis.hut.ft