The helical coordinate

Kolmogoroff spectral factorization

With Fourier analysis we find a method of spectral factorization that is as fast as Fourier transformation, namely $N\log N$ for a matrix of size $N$ . This is very appealing. An earlier version of this book included such an algorithm. Pedagogically, I didn't like it in this book because it requires lengthy off-topic discussions of Fourier analysis which are already found in both my first book FGDP and my third book PVI.

The Kolmogoroff calculation is based on the logarithm of the spectrum. The logarithm of zero is minus infinity -- an indicator that perhaps we cannot factorize a spectrum which becomes zero at any frequency. Actually, the logarithmic infinity is the gentlest kind. The logarithm of the smallest nonzero value in single precision arithmetic is about $-36$ which might not ruin your average calculation. Mathematicians have shown that the integral of the logarithm of the spectrum must be bounded so that some isolated zero values of the spectrum are not a disaster. In other words, we can factor the (negative) second derivative to get the first derivative. This suggests we will never find a causal bandpass filter. It is a contradiction to desire both causality and a spectral band of zero gain.

The weakness of the Kolmogoroff method is related to its strength. Fourier methods strictly require the matrix to be a band matrix. A problem many people would like to solve is how to handle a matrix that is ``almost'' a band matrix -- a matrix where any band changes slowly with location.

The helical coordinate