thesis

obtained by computing the characteristic polynomial of the matrix A :
d�
PA(X) = det(X × Id − A) = akX
k=0
d−k .
The values bk are obtained with bk = Tk[i, 0], where :
Tk =
k�
ak−lA l .
l=0
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We give an outline of the proof here. We start from the Cayley-Hamilton theorem,
which states that PA(A) = 0. We multiply this equation by Yn−d :
d�
ad−kA k Yn−d = 0.
k=0
We then calculate A k Yn−d by induction :
A k k�
Yn−d = Yn−d+k − A
p=1
k−p Xn−d+p
and we substitute it in the equation above, which gives :
d�
d� k�
0 = ad−kYn−d+k − ad−k A
k=0
k=0 p=1
k−p Xn−d+p.
We then obtain the desired result by looking at coordinate i.
Using this technique, electrode compensation is very fast (close to real time
with sampling rate 10 kHz), even though we implemented it in Python, an interpreted
language.