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Brief Tour of Wavelet Theory

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Wavelets in Neuroscience

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

In this chapter, the main definitions of wavelet theory are given. To explain the basic ideas of the continuous wavelet transform, we describe a transition from Fourier analysis to wavelets. Mother functions and numerical techniques for implementing the wavelet transform are described. The problem of visualising the results is considered. Finally, features of the discrete wavelet transform are discussed.

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Notes

  1. 1.

    Note that hereafter we will consider only the positive range of frequencies, since the negative frequency region is the “mirror image” of the positive one and does not provide any additional useful information.

  2. 2.

    More precisely, amplitude spectra of the Fourier transform.

  3. 3.

    Of course, in the case of experimental signals or data from numerical simulation, researchers deal with finite time series.

  4. 4.

    Here, for simplicity, a rectangular window is used. In a more general case (known as the Gabor transform), we use a window function g(t) that is localized in both the time and frequency domains.

  5. 5.

    This is the most typical case in experimental studies. However, data can be acquired in such a way that each data point is related to an arbitrary instant of time. This happens, e.g., for point processes represented by RR intervals of the electrocardiogram [31, 32]. In such a case, the relevant algorithms must be modified [31, 32].

  6. 6.

    Notice the difference between ψ, \(\hat{\psi }\) and Ψ, \(\hat{\varPsi }\) used for the continuous and discrete transforms, respectively.

  7. 7.

    Here we do not consider iterations for calculation of the Fourier image of the signal \(\hat{x}\), since this transform should be performed only once.

  8. 8.

    For time series with length N, only \(\mathcal{L}\times 2N(1 +\log _{2}N)\) arithmetic operations are needed to obtain the wavelet surface with the described technique for reducing edge effects.

  9. 9.

    The dimension of the phase space is equal to the number of quantities required to fully characterize the state of the system under study.

  10. 10.

    Of course, if one deals with a system dimension of 3 or higher.

  11. 11.

    Nevertheless, the phase synchronization of such systems can usually be detected by means of indirect indications [58, 63] and measurements [64].

  12. 12.

    For practical purposes, mother wavelets can also be constructed from tabulated segments of time series (see [30, 71]). This subject will be discussed in detail in Sect. 6.3.

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Authors and Affiliations

  1. Research and Education Center ‘Nonlinear Dynamics of Complex Systems’, Saratov State Technical University, Saratov, Russia

    Alexander E. Hramov, Alexey A. Koronovskii & Alexey N. Pavlov

  2. Department of Nonlinear Processes, Saratov State University, Saratov, Russia

    Alexander E. Hramov & Alexey A. Koronovskii

  3. Department of Applied Mathematics, Complutense University, Madrid, Spain

    Valeri A. Makarov

  4. Physics Department, Saratov State University, Saratov, Russia

    Alexey N. Pavlov

  5. Institute for Higher Nervous Activity and Neurophysiology, Russian Academy of Sciences, Moscow, Russia

    Evgenia Sitnikova

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Hramov, A.E., Koronovskii, A.A., Makarov, V.A., Pavlov, A.N., Sitnikova, E. (2015). Brief Tour of Wavelet Theory. In: Wavelets in Neuroscience. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43850-3_2

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