A simple introduction to the KLT and BAM-KLT

Part of the Springer Praxis Books book series (PRAXIS)


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.


Fast Fourier Transform Global Navigation Satellite System Global Navigation Satellite System Final Variance Analytic Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. Karhunen, “ϋber lineare Methoden in der Wahrscheinlichkeitsrechnung,” Ann. Acad. Sci. Fennicae, Series A 1, Math. Phys., 37 (1946), 3–79.Google Scholar
  2. 2.
    M. Loève, “Fonctions Aléatoires de Second Ordre,” Rev. Sci., 84(4) (1946), 195–206.MathSciNetGoogle Scholar
  3. 3.
    M. Loève, Probability Theory: Foundations, Random Sequencies, Van Nostrand, Princeton, NJ, 1955.Google Scholar
  4. 4.
    C. Maccone, Telecommunications, KLT and Relativity, Volume 1, IPI Press, Colorado Springs, CO, 1994, ISBN # 1-880930-04-8. This book embodies the results of some 30 research papers published by the author about the KLT in the 15-year span 1980–1994 in peer-reviewed journals.Google Scholar
  5. 5.
    S. Montebugnoli, C. Bortolotti, D. Caliendo, A. Cattani, N. D’Amico, A. Maccaferri, C. Maccone, J. Monari, A. Orlati, P. P. Pari et al., “SETI-Italia 2003 Status Report and First Results of a KL Transform Algorithm for ETI Signal Detection,” paper IAC-03- IAA.9.1.02 presented at the 2003 International Astronautical Congress held in Bremen, Germany, September 29–October 3, 2003.Google Scholar
  6. 6.
    F. Biraud, “SETI at the Nançay Radio-telescope,” Acta Astronautica, 10 (1983), 759–760.ADSCrossRefGoogle Scholar
  7. 7.
    C. Maccone, “Advantages of the Karhunen–Loève Transform over Fast Fourier Transform for Planetary Radar and Space Debris Detection,” Acta Astronautica, 60 (2007), 775–779.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.International Academy of Astronautics and Istituto Nazionale di AstrofisicaTorino (Turin)Italy

Personalised recommendations