sfact - discrete time spectral factorization
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 z=poly(0,'z'); p=(z-1/2)*(2-z) w=sfact(p); w*numer(horner(w,1/z)) //matrix example 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(P)) roots(det(gtild(F,'d'))) //The stable roots roots(det(F)) //The antistable roots clean(P-F*gtild(F,'d')) //Example of continuous time use s=poly(0,'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=numer(w); //take the numerator fn=sfact(wn);f=numer(horner(fn,(1-s)/(s+1))); //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)