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Transformations of Functions and Signals

  • Simon Širca
  • Martin Horvat
Part of the Graduate Texts in Physics book series (GTP)

Abstract

The discrete Fourier transformation is one of the most important tools in the analysis of functions and signals, but its detailed numerical aspects are often disregarded or neglected. Fourier and sampling theorems, Parseval’s equality, and power spectral densities are introduced, and the concepts of signal uncertainty, aliasing, and leakage are discussed, along with the caveats due to different notations and nomenclatures. Transformations with orthogonal polynomials (Legendre and Chebyshev) are described, and their relation to quadrature formulas is explained. The numerical computation of the Laplace transformation and its inverse is introduced in the context of differential equations for transfer functions of physical systems. The continuous and discrete Hilbert transformations are discussed next, introducing the concept of the analytic signal, and their relation to the Kramers–Krönig dispersion relations is given. The last Section deals with the continuous wavelet transform. The Examples and Problems include the multiplication of polynomials by FFT, the Fourier analysis of realistic acoustic signals and the Doppler effect, and the analysis of brain potentials by the Hilbert transform.

Keywords

Orthogonal Polynomial Discrete Wavelet Transform Discrete Fourier Transform Wavelet Transformation Chebyshev Polynomial 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Simon Širca
    • 1
  • Martin Horvat
    • 1
  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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