WILSON-BURG SPECTRAL FACTORIZATION

(If you are new to this material, you should pass over this section.) Spectral factorization is the job of taking a power spectrum and from it finding a causal (zero before zero time) filter with that spectrum. Methods for this task (there are many) not only produce a causal wavelet, but they typically produce one whose convolutional inverse is also causal. (This is called the ``minimum phase'' property.) In other words, with such a filter we can do stable deconvolution. Here I introduce a new method of spectral factorization that looks particularly suitable for the task at hand. I learned this new method from John Parker Burg who attributes it to an old paper by Wilson (I find Burg's explanation, below, much clearer than Wilson's.)

Below find subroutine lapfac() which was used in the previous section to factor the Laplacian operator. To invoke the factorization subroutine, you need to supply one side of an autocorrelation function. For example, let us specify the negative of the 2-D Laplacian (an autocorrelation) in a vector n = $256\times 256$ points long.

        rr(1)     =  4.
        rr(2)     = -1.
        rr(1+256) = -1.

Subroutine lapfac() finds the helical derivative (factored negative Laplacian) and then prepares the required filter coefficient tables for the helix convolution and deconvolution subroutines. lapfacfactor 2-D Laplacian Subroutine lapfacn() has its main job done by subroutine wilson_factor() shown after the Wilson-Burg theory.

 

• Wilson-Burg theory