I’m defining a number system where numbers form a polynomial ring with cyclic convolution as multiplication.
My DSP math is a bit rusty so I’m asking when does inverse circular convolution exist? You can easily calculate it using FFT but I’m uncertain when does it exist? I would like to have a full number system where each number only has a single well defined inverse. Another part of my problem is derivation. Let c be number in my number system C[X] where coefficients are complex numbers. Linear functions can be derivated easily but I’m struggling to minimize mean squared error (i = 0..degree(C[X]), s_i(c) selects i:th coefficient of the number, s_i(x): C[x]->C):
error(W) = Exy{sum(i)(0.5|s_i(Wx - y)|^2)}
I can solve problem in case of complex numbers W E C but not in case of W E C[X] where multiplication is circular convolution. In practice my linear neural network code diverges to infinity when I try to minimize squared error.
Pointing any online materials that would help would be great.
The subject you describe sounds like circulants. They are matrices whose product rule is equivalent to multiplication of vectors through convolution. I can give you a lot more information on them since I'm probably one of the world experts on them. I'd be willing to send you my book if you are interested.
By the way, what is DSP?
--- Alun
Have you tried Mathematica? It's my favorite software for Maths.
Better than Python (easier to use).
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