**Fourier Spectrum:**

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

where

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

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

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

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

where

- is the real part of a complex number;
- 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 .