Signal Separation

Abstract

Now we address some of the problems associated with the first task in Table 1.1. Given observations contaminated by other sources, how do we clean up the signal of interest so we can perform the analysis for Lyapunov exponents, dimensions, model building, etc.? In linear analysis the problem concerns the extraction of sharp, narrowband linear signals from broadband “noise”. This is best done in Fourier domain, but that is not the working space for nonlinear dynamics. The problem of separating nonlinear signals from one another is done in the time domain of dynamical phase space. To succeed in signal separation, we need to characterize one or all of the superposed signals in some fashion which allows us to differentiate them. This is precisely what we do in linear problems as well, though the distinction is spectral there. If the observed signal s(n) is a sum of the signal we want, call it s₁(n), and other signals s₂(n), s₃(n),...,

$$s{\text{(}}n{\text{) = }}{s^{\text{1}}}{\text{ (}}n{\text{) + }}{s^{\text{2}}}{\text{ (}}n{\text{) + }}{s^{\text{3}}}{\text{ (}}n{\text{) + }}...{\text{,}}$$

(7.1)

then we must identify some distinguishing characteristic of s₁(n) which either the individual s_i(n); i > 1 or the sum does not possess.