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Fourier Spectrum

Fourier Spectrum:

Any continuous function Math image defined on a finite interval Math image of length Math image can be represented as a weighted sum of cosine functions with periods Math image :

Glossary Math

where

  • Math image is the frequency of the i-th Fourier component;
  • Math image is the amplitude of the i-th component;
  • Math image is the phase of the i-th component.

The function Math image describing the dependence of the amplitude Math image on the frequency Math image in the above expression is called the amplitude spectrum of the function Math image .

The function Math image describing the dependence of the phase Math image on the frequency Math image is called the phase spectrum of the function Math image .

Thus, the Fourier spectrum of a function Math image is represented by two functions of the frequency Math image - the amplitude spectrum Math image and the phase spectrum Math image . These two are often combined into a single complex-valued function Math image :

Glossary Math

where

  • Math image is the real part of a complex number;
  • Math image is the imaginable part of a complex number;

The concept of the spectrum plays an important role in signal processing , time series analysis , spectral analysis .

See also: power spectrum .

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