Re: Spectral Matrix Factorization via Wilson Method

On Aug 29, 7:43 am, Mullen....@xxxxxxxxx wrote:
Hi DSP gurus,

I am working on an application of Tunnicliffe Wilson's method for
inner-outer factorization of a spectral density matrix, S(f), (where
S(f) is an q x q matrix of auto- and cross-spectra at freq f between q
channels) into minimum-phase factor Y(z) such that S(f)=Y(z)Y(z)*
(where z=exp(jf) and * = matrix adjoint). I've been attempting to
implement Wilson's method as described in his 1972 paper "The
Factorization of Matricial Spectral Densities, SIAN J. Appl. Math.
Vol. 23, No. 4, December 1972." This method obtains the solution
using the Newton-Raphson method to solve a system of quadratic
equations.

First of all let me mention that I'm not a signal processing expert by
any means (my field is cognitive science / comp. neuroscience),
although I'm familiar with some dsp terminology and many of the
statistical methods frequently used in spectral analysis. So while I
can probably make sense of most of what you guys might say, overly
complex terminology will likely go over my head...

I've attempted to replicate Wilson's method in MATLAB, but the
algorithm does not quite converge on the correct solution. Perhaps one
issue (a very simple one, I'm sure), which I'm not 100% sure of, is
the initial conditions. Wilson proved that the algorithm will
converge as long as the initial guess for Y(z) is nonsingular and
positive in the closed unit disk. Forgive my ignorance, but does this
mean that any invertible, positive matrix would work? i.e. in the q=2
case:

for all f, Y_0(f) = [0.3 0.4
0.1 0.2]

Also, Wilson notes that the algorithm only proveably converges
quadratically if S(f)>0; however, cross-spectra often contain negative
power over wide frequency ranges. I'm curious how this would affect
the performance of this algorithm in the q>=2 case.

If anyone reading this is familiar with Wilson's method and could help
me in understanding why my algorithm does not converge, I'd very much
appreciate it. The method itself does not appear excessively
complicated, and it seems close to working properly, but something is
missing. (In the q=1 case (factoring auto-spectrum of 1 series) the
real part converges to the true solution, but the imaginary part does
not, for q>=2 it somewhat resembles the correct solution, but is still
clearly off). I have MATLAB code which I could provide, if you had the
time to help me out.

Alternately, if anyone here already has code in any programming
language which does the factorization for a given spectral density
matrix (*not* a matrix of auto/cross-covariance coefficients) and
would be willing to share, that would be phenomenal as Wilson's method
is just a small component in a much larger algorithm for figuring out
information flow in the brain (but I've been stuck on it for a
while).

Thanks!

D. C Youla, “On the factorization of rational matrices,” IRE Trans.
Inform.
F. R. Gantmacher, The theory of Matrices, voL 1. New York: Chelsea,
1959

You need to reduce it to its Smith Normal form.
.

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