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Scilab Help >> Polynomials > sfact


discrete time spectral factorization





real polynomial matrix


Finds F, a spectral factor of P. 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

// matrix example
z = %z;
F1 = [z-1/2, z+1/2, z^2+2; 1, z, -z; z^3+2*z, z, 1/2-z];
P = F1*gtild(F1,'d');  // P is symmetric
F = sfact(P)
roots(det(gtild(F,'d')))  //The stable roots
roots(det(F))             //The antistable roots

// 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)

See also

  • gtild — tilde operation
  • fspecg — stable factorization of continuous time dynamical systems
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