Advertisement

Wavelets

  • Walter J. Freeman
  • Rodrigo Quian Quiroga
Chapter

Abstract

In the previous chapter, we showed that a key issue with the short-time Fourier transform (STFT) is the choice of the window length, given the basic limitation imposed by the uncertainty principle of signal analysis. Short windows give good time (but bad frequency) resolution, and conversely, long windows give good frequency (but bad time) resolution (see Sect. 3.3). In the late 1970s, Jean Morlet, a geophysicist working for a French oil company, realized that the STFT was not suitable for the study of his seismic data. He observed that a good compromise between time and frequency resolution was not possible because high-frequency patterns had a shorter duration compared to the low-frequency ones. So, a single window for all frequencies would not do. His solution was quite straightforward: he just took different window sizes for different frequencies, or more accurately, he took a cosine function tapered with a Gaussian (a Gabor function, see Sect. 3.2) and compressed it or stretched it in time to get the different frequencies (see Fig. 4.1). Then, instead of always having the same window size, he had the same wave shape at different scales, that is, with a variable size. With this simple trick, he created the basis of wavelets!

Keywords

Wavelet Coefficient Digital Filter Wavelet Function Mother Wavelet Haar Wavelet 
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.

Supplementary material

References

  1. Chui C (1992) An introduction to wavelets. Academic, San DiegoGoogle Scholar
  2. Clark I, Biscay R, Echeverria M, Virues T (1995) Multiresolution decomposition of non-stationary EEG signals: a preliminary study. Compt Biol Med 25:373–382CrossRefGoogle Scholar
  3. Cohen L (1995) Time-frequency analysis. Prentice-Hall, New JerseyGoogle Scholar
  4. D’attellis C, Isaacson S, Sirne R (1997) Detection of epileptic events in electroencephalograms using wavelet analysis. Ann Biomed Eng 25:286–293PubMedCrossRefGoogle Scholar
  5. Daubechies I (1996) Where do wavelets come from? – A personal point of view. Proc IEEE 84:510–513CrossRefGoogle Scholar
  6. Grossmann A, Morlet J (1984) Decomposition of Hardy Functions into square integrable wavelets of constant shape. SIAM J Math Anal 15:723–736CrossRefGoogle Scholar
  7. Kiymik MK, Akin M, Subasi A (2004) Automatic recognition of alertness level by using wavelet transform and artificial neural network. J Neurosci Methods 139:231–240PubMedCrossRefGoogle Scholar
  8. Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Analysis Machine Intell 2:674–693CrossRefGoogle Scholar
  9. Mallat S (1998) A wavelet tour of signal processing. Academic, San DiegoGoogle Scholar
  10. Quian Quiroga R. (1998). Quantitative analysis of EEG signals: Time-frequency methods and chaos theory. PhD thesis. University of Lubeck, Germany.Google Scholar
  11. Quian Quiroga R, Schürmann M (1999) Functions and sources of event-related EEG alpha oscillations studied with the Wavelet Transform. Clin Neurophysiol 110:643–655CrossRefGoogle Scholar
  12. Quian Quiroga R, Sakowicz O, Basar E, Schurmann M (2001) Wavelet transform in the analysis of the frequency composition of evoked potentials. Brain Res Protoc 8:16–24CrossRefGoogle Scholar
  13. Quian Quiroga R, Kraskov A, Kreuz T, Grassberger P (2002) Performance of different synchronization measures in real data: A case study on electroencephalographic signals. Phys Rev E 65:041903CrossRefGoogle Scholar
  14. Samar VJ, Swartz KP (1995) Multiresolution analysis of event-related potentials by wavelet decomposition. Brain Cogn 27:398–438PubMedCrossRefGoogle Scholar
  15. Samar VJ, Bopardikar A, Rao R, Swartz K (1999) Wavelet analysis of neuroelectric waveforms: A conceptual tutorial. Brain Lang 66:7–60PubMedCrossRefGoogle Scholar
  16. Sartoretto F, Ermani M (1999) Automatic detection of epileptiform activity by single-level wavelet analysis. Clin Neurophysiol 110:239–249PubMedCrossRefGoogle Scholar
  17. Schiff S, Aldroubi A, Unser M, Sato S (1994a) Fast wavelet transformation of EEG. Electr Clin Neurophysiol 91:442–455CrossRefGoogle Scholar
  18. Schiff S, Milton J, Heller J, Weinstein S (1994b) Wavelet transforms and surrogate data for electroencephalographic spike and seizure localization. Optical Eng 33:2162–2169CrossRefGoogle Scholar
  19. Senhadji L, Dillenseger JL, Wendling F, Rocha C, Linie A (1995) Wavelet analysis of EEG for three-dimensional mapping of epileptic events. Annals of Biom Eng 23:543–552CrossRefGoogle Scholar
  20. Strang G, Nguyen T (1996) Wavelets and filter banks. Wellesley-Cambridge, WellesleyGoogle Scholar
  21. Unser M, Aldroubi A (1996) A review of wavelets in biomedical applications. Proc IEEE 84:626–638CrossRefGoogle Scholar
  22. Unser M, Aldroubi A, Eden M (1992) On the asymptotic convergence of B-Spline wavelets to Gabor functions. IEEE Trans Inf Theory 38:864–872CrossRefGoogle Scholar
  23. Yeung N, Bogacz R, Holroyd CB, Cohen JC (2004) Detection of synchronized oscillations in the electroencephalogram: An evaluation of methods. Psychophysiology 41:822–832PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Walter J. Freeman
    • 1
  • Rodrigo Quian Quiroga
    • 2
  1. 1.Molecular and Cell BiologyUniversity of California at BerkeleyBerkeleyUSA
  2. 2.Centre for Systems NeuroscienceUniversity of LeicesterLeicesterUK

Personalised recommendations