Spectral factorization

We can now extend the equations derived for real numbers to polynomials of Z, with $Z=e^{i\omega t}$, and obtain spectral factorization algorithms similar to the Wilson-Burg method Sava et al. (1998), as follows:  

$$\fbox {$ \displaystyle X_{n+1}=\frac{S+X_{n} \bar{G}}{ \bar{X_n} + \bar{G}} $}$$ (8)

If L represents the limit of the series in (8),

$$L \bar{L} + L \bar{G} = S + L \bar{G}$$

and so

$$L \bar{L} = S$$

Therefore, L represents the causal or anticausal part of the given spectrum $S=X\bar{X}$.

Table 3 summarizes the spectral factorization relationships equivalent to those established for real numbers in Table 1.

   

$$X_{n+1} =\frac{S+X_n \bar G }{ \bar X_n+\bar G }$$ General

$$X_{n+1} =\frac{S+X_n }{ \bar X_n+1 }$$ Muir

$$X_{n+1} =\frac{S+X_n \bar X_{n-1}}{ \bar X_n+\bar X_{n-1}}$$ Secant

$$X_{n+1} =\frac{S+X_n \bar X_n }{2\bar X_n }$$ Newton

$$X_{n+1} =\frac{S+X_n\sqrt{S} }{ \bar X_n+\sqrt{S} }$$ Ideal

The convergence properties are similar to those derived for real numbers. As shown above, the Newton-Raphson method should have the fastest convergence.