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

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. 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.


Fast Fourier Transform Fourier Spectrum Final Variance Wide Bandwidth Narrow Bandwidth 
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.


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  1. [1]
    K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung,” Ann. Acad. Sci. Fennicae, Series A 1, Math. Phys., 37 (1946), 3–79.MathSciNetGoogle 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.CrossRefADSGoogle 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.CrossRefADSGoogle Scholar

Annotated Bibliography

  1. [8]
    C. Maccone, “Special Relativity and the Karhunen—Loève Expansion of Brownian Motion,” Nuovo Cimento, Series B, 100 (1987), 329–342.MathSciNetGoogle Scholar
  2. [9]
    C. Maccone, “Eigenfunctions and Energy for Time-Rescaled Gaussian Processes,” Bollettino dell’Unione Matematica Italiana, Series 6, 3-A (1984), 213–219MathSciNetGoogle Scholar
  3. [10]
    C. Maccone, “The Time-Rescaled Brownian Motion B(t 2H),” Bollettino dell’Unione Matematica Italiana, Series 6, 4-C (1985), 363–378; C. Maccone, “The Karhunen—Loève Expansion of the Zero-Mean Square Process of a Time-Rescaled Gaussian Process,” Bollettino dell’Unione Matematica Italiana, Series 7, 2-A (1988), 221–229.MathSciNetGoogle Scholar
  4. [12]
    C. Maccone, “Relativistic Interstellar Flight and Genetics,” Journal of the British Interplanetary Society, 43 (1990), 569–572.Google Scholar
  5. [13]
    C. Maccone, “Relativistic Interstellar Flight and Gaussian Noise,” Acta Astronautica, 17(9) (1988), 1019–1027.CrossRefADSGoogle Scholar
  6. [14]
    C. Maccone, “Relativistic Interstellar Flight and Instantaneous Noise Energy,” Acta Astronautica, 21(3) (1990), 155–159.CrossRefADSMathSciNetGoogle Scholar
  7. [15]
    C. Maccone, “The Data Compression Problem for the ‘Gaia’ Astrometric Satellite of ESA,” Acta Astronautica, 44(7–12) (1999), 375–384.CrossRefADSGoogle Scholar
  8. [16]
    R. S. Dixon, and M. Klein, “On the detection of unknown signals,” Proceedings of the Third Decennial US-USSR Conference on SETI held at the University of California at Santa Cruz, August 5–9, 1991. Later published in the Astronomical Society of the Pacific (ASP) Conference Series (Seth Shostak, Ed.), 47 (1993), 128–140.Google Scholar
  9. [17]
    C. Maccone, Karhunen-Loève versus Fourier Transform for SETI, Lecture Notes in Physics, Springer-Verlag, Vol. 390 (1990), pp. 247–253. These are the Proceedings (J. Heidmann and M. Klein, Eds.) of the Third Bioastronomy Conference held in Val Cenis, Savoie, France, June 18–23, 1990.Google Scholar
  10. [18]
    R. Eckers, K. Cullers, J. Billingham, and L. Scheffer (Eds.), SETI 2020, SETI Institute, Mountain View, CA, 2002, p. 234, note 13. The authors say: “Currently (2002) only the Karhunen Loeve (KL) transform [Mac94] shows potential for recognizing the difference between incidental radiation technology and white noise. The KL transform is too computationally intensive for the present generation of systems. The capability for using the KL transform should be added to future systems when computational requirements become affordable.”Google Scholar
  11. [19]
    C. Maccone, “The Karhunen—Loève Transform: A Better Tool than the Fourier Transform for SETI and Relativity,” Journal of the British Interplanetary Society, 47 (1994), 1.ADSGoogle Scholar
  12. [20]
    S. Montebugnoli, and C. Maccone, “SETI-Italia Status Report 2001”, a paper presented at the 2001 IAF Conference held in Toulouse, France, October 1–5, 2001.Google Scholar
  13. [21]
    A. K. Jain, “A Fast Karhunen—Loève Transform for a Class of Random Processes,” IEEE Trans. Commun., COM-24 (1976), 1023–1029.CrossRefGoogle Scholar
  14. [22]
    F. Schillirò, S. Pluchino, C. Maccone, and S. Montebugnoli, La KL Transform: considerazioni generali sulle metodologie di analisi ed impiego nel campo della Radioastronomia, Istituto Nazionale di Astrofisica (INAF)/Istituto di Radioastronomia (IRA), Technical Report, January 2007 [in Italian].Google Scholar
  15. [23]
    C. Maccone, “Innovative SETI by the KLT,” Proceedings of the Bursts, Pulses and Flickering Conference held at Kerastari, Greece, June 13–18, 2007. Available at POS (Proceedings of Science) website Google Scholar
  16. [24]
    S. Yatawatta, pers. commun., June 17, 2008.Google Scholar
  17. [25]
    C. Maccone, “Relativistic Optimized Link by KLT,” Journal of the British Interplanetary Society, 59 (2006), 94–98.Google Scholar

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© Praxis Publishing Ltd, Chichester, UK 2009

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