A simple introduction to the KLT (Karhunen—Loève Transform)

Abstract

This chapter is a simple introduction about using the Karhunen—Loève Transform (KLT) to extract weak signals from noise of any kind. In general, the noise may be colored and over wide bandwidths, and not just white and over narrow bandwidths. We show that the signal extraction can be achieved by the KLT more accurately than by the Fast Fourier Transform (FFT), especially if the signals buried into the noise are very weak, in which case the FFT fails. This superior performance of the KLT happens because the KLT of any stochastic process (both stationary and non-stationary) is defined from the start over a finite time span ranging between 0 and a final and finite instant T (contrary to the FFT, which is defined over an infinite time span). We then show mathematically that the series of all the eigenvalues of the autocorrelation of the (noise + signal) may be differentiated with respect to T yielding the “Final Variance” of the stochastic process X(t) in terms of a sum of the first-order derivatives of the eigenvalues with respect to T. Finally, we prove that this new result leads to the immediate reconstruction of a signal buried into the thick noise. We have thus put on a strong mathematical foundation a set of very important practical formulae that can be applied to improve SETI, the detection of exoplanets, asteroidal radar, and also other fields of knowledge like economics, genetics, biomedicine, etc. to which the KLT can be equally well applied with success.