Definitions
Fourier transform
The Fourier transform (or Fourier spectrum, or frequency spectrum) of a complex function g of 2 independent variables x and y is noted
or FT(g), and is defined by:
is a function, taking complex values, of two independent variables u and v. Those variables are considered as spatial frequencies.
In a similar fashion, the inverse FT of a function
is noted
and is defined by:
The functions FT and FT-1 only differ by the sign of the exponent.
The Fourier spectrum
of a function g is therefore simply the ensemble of weighting factors that should be applied to each elementary function
in order to restore the function g.
We will come back to the physical meaning of these elementary functions later in the document.
Dirac delta distribution
The Dirac delta distribution (also simply called delta function) δ(x,y) is defined by its effect when used inside an integral. Mathematically, we have:
As a simple representation of this “function”, we can consider that it takes an infinite value at the point {x=0, y=0}, it takes zero values everywhere else (it is defined for all real numbers), and it verifies
.