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.
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;
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 .