The Fourier Transform, named in honor of the French mathematician and physicist Jean Baptiste Joseph Fourier (1768-1830), who in 1807 claimed that any function could be expanded as a series of sinusoidal waves of appropriate amplitude and phase in his "Théorie analytique de la chaleur" (published 1822.)

The main idea behind the Fourier transform is to represent a periodic function, , as an infinite series of trigonometric functions, such that

or in exponential form:

where is the interval for which makes one period, and are called Fourier coefficients, is called the fundamental frequency and the coefficients are complex.

Although Fourier claims where not entirely true for all continuous functions, it does a perfect representation of discrete signals. In this case, the summation takes place a finite number of times and the Fourier transform, which is then termed the discrete Fourier transform, is said to have such a degree:

The Fourier coefficients for a discrete signal can be found as follows:

Because the Fourier transform breaks a function into its set of amplitudes and phases for the sine and cosine functions at a given number of frequencies, it is said that the Fourier transform decomposes a signal into its frequency components, hence the terminology frequency domain representation. In this context, the power of a discrete signal at any frequency multiple is given by: