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discrete time spectral factorization
real polynomial matrix
F, a spectral factor of
P is a polynomial matrix such that
each root of
P has a mirror image w.r.t the unit
circle. Problem is singular if a root is on the unit circle.
sfact(P) returns a polynomial matrix
F(z) which is antistable and such that
P = F(z)* F(1/z) *z^n
For scalar polynomials a specific algorithm is implemented. Algorithms are adapted from Kucera's book.
// Simple polynomial example p = (%z -1/2) * (2 - %z) w = sfact(p); w*horner(w, 1/%z).num
// Example of continuous time use s = %s; p = -3*(s+(1+%i))*(s+(1-%i))*(s+0.5)*(s-0.5)*(s-(1+%i))*(s-(1-%i)); p = real(p); // p(s) = polynomial in s^2 , looks for stable f such that p=f(s)*f(-s) w = horner(p,(1-s)/(1+s)); // bilinear transform w=p((1-s)/(1+s)) wn = w.num; // take the numerator fn = sfact(wn); f = horner(fn,(1-s)/(s+1)).num; // Factor and back transform f = f/sqrt(horner(f*gtild(f,'c'),0)); f = f*sqrt(horner(p,0)); // normalization roots(f) // f is stable clean(f*gtild(f,'c')-p) // f(s)*f(-s) is p(s)
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