Skip to main content

Fourier Analysis of Signals

Abstract

As we have seen in the last chapter, music signals are generally complex sound mixtures that consist of a multitude of different sound components. Because of this complexity, the extraction of musically relevant information from a waveform constitutes a difficult problem. A first step in better understanding a given signal is to decompose it into building blocks that are more accessible for the subsequent processing steps. In the case that these building blocks consist of sinusoidal functions, such a process is also called Fourier analysis. Sinusoidal functions are special in the sense that they possess an explicit physical meaning in terms of frequency. As a consequence, the resulting decomposition unfolds the frequency spectrum of the signal—similar to a prism that can be used to break light up into its constituent spectral colors. The Fourier transform converts a signal that depends on time into a representation that depends on frequency. Being one of the most important tools in signal processing, we will encounter the Fourier transform in a variety of music processing tasks.

Keywords

  • Fast Fourier Transform
  • Discrete Fourier Transform
  • Original Signal
  • Analog Signal
  • Window Function

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-21945-5_2
  • Chapter length: 76 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-21945-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. L. BUTZER, W. SPLETTST ÖSSER, AND R. L. STENS, The sampling theorem and linear prediction in signal analysis, Jahresbericht der Deutschen Mathematiker-Vereinigung, 90 (1988), pp. 1–70.

    Google Scholar 

  2. M. CLAUSEN AND U. BAUM, Fast Fourier Transforms, BI Wissenschaftsverlag, 1993.

    Google Scholar 

  3. J. W. COOLEY AND J. W. TUKEY, An algorithm for the machine calculation of complex Fourier series, Mathematics of Computation, 19 (1965), pp. 297–301.

    Google Scholar 

  4. I. DAUBECHIES, Ten lectures on wavelets, Society for Industrial and Applied Mathematics (SIAM), 1992.

    Google Scholar 

  5. G. B. FOLLAND, Real Analysis, John Wiley & Sons, 1984.

    Google Scholar 

  6. D. GABOR, Theory of communication, Journal of the Institution of Electrical Engineers (IEE), 93 (1946), pp. 429–457.

    Google Scholar 

  7. F. J. HARRIS, On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE, 66 (1978), pp. 51–83.

    Google Scholar 

  8. J. R. HIGGINS, Five short stories about the cardinal series, Bulletin of the American Mathematical Society, 12 (1985), pp. 45–89.

    Google Scholar 

  9. B. B. HUBBARD, The world according to wavelets, AK Peters, Wellesley, Massachusetts, 1996.

    Google Scholar 

  10. G. KAISER, A Friendly Guide to Wavelets, Modern Birkhäuser Classics, 2011.

    Google Scholar 

  11. MATLAB, High-performance numeric computation and visualization software. The Math-Works Inc., http://www.mathworks.com, 2013.

  12. M. MÜLLER, Information Retrieval for Music and Motion, Springer Verlag, 2007.

    Google Scholar 

  13. A. V. OPPENHEIM, A. S. WILLSKY, AND H. NAWAB, Signals and Systems, Prentice Hall, 1996.

    Google Scholar 

  14. J. G. PROAKIS AND D. G. MANOLAKIS, Digital Signal Processing, Prentice Hall, 1996.

    Google Scholar 

  15. G. STRANG AND T. NGUYEN, Wavelets and Filter Banks, Wellesley-Cambridge Press, 2nd ed., 1996.

    Google Scholar 

  16. M. VETTERLI, J. KOVACEVIC, AND V. K. GOYAL, Foundations of Signal Processing, Cambridge University Press, http://fourierandwavelets.org/, 1st ed., 2013.

  17. ------, Fourier and Wavelet Signal Processing, Cambridge University Press, http://fourierandwavelets.org/, 1st ed., 2014.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Meinard Müller .

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Müller, M. (2015). Fourier Analysis of Signals. In: Fundamentals of Music Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-21945-5_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21945-5_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21944-8

  • Online ISBN: 978-3-319-21945-5

  • eBook Packages: Computer ScienceComputer Science (R0)