# Fourier Spectrum

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 .

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