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Let us compute the real circular convolution of dimension N=a,n, with "n" and "a" respectively prime, on two numerical series (P(i)) and (Q(i)).
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Algebraic Techniques For Computing Real Monodimensional Circular
Let us compute the real circular convolution of dimension N=a,n, with "n" and
"a" respectively prime, on two numerical series (P(i)) and (Q(i)).
For instance, let us choose a = 4. Then, both series will first be converted
into matrices of four lines and n columns each, by index rearrangements; i.e.,
(P(i)) and (Q(i)) are converted into
The final output terms w(i,j) are then obtained by combining the u(i,j) and
v(i,j), so that:
This process may be extended to circular convolutions of dimension N = a.n
by reducing Z -1 into irreducible factors, splitting the process as we did above,
and recombining the results by using the Chinese remainder theorem.