The KLT (Karhunen–Loève Transform) to extend SETI searches to broad-band and extremely feeble signals

Abstract

The KLT (acronym for Karhunen–Loève Transform) is a mathematical algorithm superior to the classical FFT in many regards: 1) The KLT can filter signals out of the background noise over both wide and narrow bands. This is in sharp contrast to the FFT that rigorously applies to narrow-band signals only. 2) The KLT can be applied to random functions that are non-stationary in time, i.e. whose autocorrelation is a function of the two independent variables t 1 and t 2 separately. Again, this is a sheer advantage of the KLT over the FFT, inasmuch as the FFT rigorously applies to stationary processes only, i.e. processes whose autocorrelation is a function of the absolute value of the difference of t 1 and t 2 only. 3) The KLT can detect signals embedded in noise to unbelievably small values of the Signal-to-Noise Ratio (SNR), like 10 −3 or so. This particular feature of the KLT is studied in detail in this paper. An excellent filtering algorithm like the KLT, however, comes with a cost that one must be ready to pay for especially in SETI: its computational burden is much higher than for the FFT. In fact, it can be shown that no fast KLT transform can possibly exist and, for an autocorrelation matrix of size N, the calculations must be of the order of N 2, rather than N log( N). Nevertheless, for moderate values of N (in the hundreds), the KLT dominates over the FFT, as shown by the numerical simulations. Finally, an important and recent (2007–2008) development in the KLT theory, called the “Bordered Autocorrelation Method” (BAM), is presented. This BAM-KLT method gets around the difficulty of the N 2 brunt calculations and ends up in the following unexpected theorem: the KLT of a feeble sinusoidal carrier embedded into a lot of white stationary noise is given by the Fourier transform of the derivative of the largest KLT eigenvalue with respect to the bordering index. This basic result is fully proved analytically in the final sections of this paper by virtue of a new theorem discovered by this author in May 2007 and called “The Final Variance Theorem”.