In ERP/P300 signal analysis, xDAWN is well-known to find the spatial filter.
I have read several reference papers about xDAWN, such as
xDAWN Algorithm to Enhance Evoked Potentials: Application to Brain–Computer Interface
A Tutorial on EEG Signal Processing Techniques for Mental State Recognition in Brain-Computer Interfaces
But I still do not know very well about xDAWN. So far, I know that the first column of D is 0 except the positions of stimuli onset, but how about the other columns? or we do not need to know the others then we can create the Toeplitz matrix?
Would you please give me an example? Where can I find the source code of xDAWN to let me study more about it?
This question is old but it is still unanswered, so here I go...
The Toeplitz matrix is just the beginning of the algorithm, and these are just matrices where all the values from the same diagonal have the same constant value. In this case, as you will have a 1 in the first column of the matrix, in the k-th row, every other column will have to have a one on that only diagonal, zero elsewhere. Think of it as the identity matrix "pushed" downwards.
They used it to represent the time-locked delay of the ERP, i.e. how many sample points do you need to wait since the onset of the stimulus until you actually start seeing the ERP. Because you multiply this matrix with A, which is the ERP signal, the result is the A matrix, representing the ERP, pushed also downwards and delayed k units.
I like the idea of "evoked potential algebra" that the authors use in the paper (Article xDAWN Algorithm to Enhance Evoked Potentials: Application to...
)Page 138 of Lotte's Brain Computer Interfaces 1 book offers a brief explanation of the algorithm that perhaps may help you as well (optimization).
You can find an implementation in OpenVibe's source code and also in the MNE-Python package.
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