Spectral factorization constructs a minimum-phase signal from its spectrum. The algorithm, suggested by Wilson (1969), approaches this problem directly with Newton's iterative method. In a Z-transform notation, Wilson's method implies solving the equation
Burg (1998, personal communication) recognized that dividing both sides of equation () by leads to a particularly convenient form, where the terms on the left are symmetric, and the two terms on the right are correspondingly strictly causal and anticausal:
Equation () leads to the Wilson-Burg algorithm, which accomplishes spectral factorization by a recursive application of convolution (polynomial multiplication) and deconvolution (polynomial division). The algorithm proceeds as follows:
An example of the Wilson-Burg convergence is shown in Table on a simple 1-D signal. The autocorrelation S(Z) in this case is , and the corresponding minimum-phase signal is A(Z) = (2+Z)(3+Z)(4+Z) = 24 + 26 Z + 9 Z2 + Z3. A quadratic rate of convergence is visible from the table. The convergence slows down for signals whose polynomial roots are close to the unit circle Wilson (1969).