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Hello

I am currently working on a project about blind source separation. I have 
read the paper "A Blind Source Separation Technique Using Second-Order 
Statistics" several times. (http://kom.aau.dk/~jash02/SOBI.pdf)

I am writing to this mailing list because I have a hard time understanding 
what the whitening does in a geometrically sense. As I understand, the 
matrix A does nothing else but transforming a vector in one space to another 
vector in another space. Since the column vectors in A are arbitrary there 
is no guarantee that the column vectors in A are orthonormal or at least 
orthogonal. I have a "feeling" that by finding a whitening matrix W the 
column vectors in WA are orthogonal or orthonormal. Is this right? In that 
case WA is just a rotation of the source vector?

I am also wondering how the "formula" for the whitening matrix W is derived 
? In the paper I do not see the steps that leads to the "formula" of W.

I hope somebody can help me with an answer?

Thanks,

Jakob

Paper: http://kom.aau.dk/~jash02/SOBI.pdf 

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"Bamse" <b...@kyllingen.dk> wrote in message news:<ckjjal$1p9q$1...@news.cybercity.dk>...
> Hello
> 
> I am currently working on a project about blind source separation. I have 
> read the paper "A Blind Source Separation Technique Using Second-Order 
> Statistics" several times. (http://kom.aau.dk/~jash02/SOBI.pdf)
> 
> I am writing to this mailing list because I have a hard time understanding 
> what the whitening does in a geometrically sense. As I understand, the 
> matrix A does nothing else but transforming a vector in one space to another 
> vector in another space. Since the column vectors in A are arbitrary there 
> is no guarantee that the column vectors in A are orthonormal or at least 
> orthogonal.

The matrix A is the mixing matrix that contains information about signal 
power as well as array calibration parameters. These parameters are in 
general not known for any given scenario/snapshot, which is why one 
speaks of "blind" source separation. If all such parameters were known 
a priori, the problem reduces to a standard beamforming problem.

> I have a "feeling" that by finding a whitening matrix W the 
> column vectors in WA are orthogonal or orthonormal. Is this right? 

I don't remember, it's many years since I read this paper. What happens, 
is that the matrix W somehow "undoes" the effects of A, so that the 
transformed problem is somewhat easier to deal with than the original 
problem.

> In that 
> case WA is just a rotation of the source vector?

Yep.

> I am also wondering how the "formula" for the whitening matrix W is derived 
> ? In the paper I do not see the steps that leads to the "formula" of W.

Well, I did read the paper just after it came out. At that time, the 
general disposition made sense (I did not dive into the fine details, 
though). If you have problems with this, I suggest you read up on some 
linear algebra. Householder transforms and Givens rotations would 
probably be very useful to you. 

A more immediate hint is that there is no such thing as a free lunch. 
The achievements in the blind source separation problem come at a price. 
Try and find out what this price is (i.e. what the conditions are on 
the problem formulation) and how the algorithm takes advantage of these
prior conditions.

> I hope somebody can help me with an answer?
> 
> Thanks,
> 
> Jakob
> 
> Paper: http://kom.aau.dk/~jash02/SOBI.pdf

Rune

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