Theorems relating to Fourier transform
Introduction
The theorems below will be frequently used as they can greatly facilitate the search of solutions to Fourier analysis problems. Let's define :
Linearity theorem
The Fourier transform of a sum of 2 functions is the sum of the 2 respective Fourier transforms:
Scaling theorem
A dilation of the spatial coordinates (x,y) results in a contraction of the frequencies u and v, and in a change in the amplitude of the whole spectrum:
Transposition, conjugation and derivation
The transpose of g(x,y) is g(-x,-y) :
.
Conjugation :
.
Derivation
:
Translation theorem
A translation in the spatial domain results in a linear phase shift in the frequency domain.
Parseval's theorem
This theorem expresses energy conservation.
Convolution theorem
The convolution of 2 functions in the spatial domain is equivalent to a simple multiplication of their Fourier transforms in the frequency domain:
Autocorrelation theorem
This is a particular case of the previous theorem:
Reciprocity theorem
Applying successively a FT and a FT-1 to a function restores that function, except at the points of discontinuity: