Scattering and diffraction. The Fourier transform  The Fourier transform, named after Jean Baptiste Joseph Fourier (French mathematician who lived between 1768 and 1830), is an almost magical mathematical tool that decomposes any periodic function of time (or periodic in space) into a sum of sinusoidal basis functions (frequency dependent), similarly to how a musical chord can be expressed as the amplitude (loudness) of its constituent notes.

T
he term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to the corresponding function of time (or space). C major chord (=C+E+G)

A good example is what the human ear makes as it receives an audio waveform and transforms it, breaking it down into different frequencies (which is what ultimately is heard). The human ear will perceive different frequencies as time passes, however, the Fourier transform contains all frequencies in the time during which the signal exists; i.e., through the Fourier transform of a function of time we get a unique frequency spectrum for the whole function. In short, the Fourier transform of a periodic function over time, is basically the frequency spectrum of that function. Fourier transform between two functions.

Function f(x), (1), is a time dependent function (red line). It is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series

The Fourier transform,ˆf(ω)(2), (blue line), which depicts amplitude vs. frequency, reveals the 6 frequencies and their amplitudes.

(1) (2) Function (2) is the Fourier transform of (1)
Function (1) is the inverse Fourier transform of (2)

Each of these basis functions into which a function can decompose, is a complex exponential of a different frequency (ω). The Fourier transform therefore gives us a unique way of viewing any function as the sum of simple sinusoids.

The inverse function of a Fourier transform is known as inverse Fourier transform (also called Fourier synthesis, or Fourier series). This inverse Fourier transform is the function that combines the contributions of all the different frequencies to recover the original function, the one that is periodic in time (or in space). The first one who applied the properties of the Fourier transform to the experiments of X-ray diffraction in the crystal was W.H. Bragg in an article published in 1915 (Phil. Trans., A, 215, 253-274).

In another chapter of these pages we will see how the Fourier transform is used to convert the so-called direct space, ie the space where are the atoms and molecules are (the function of electron density in crystals), into the "reciprocal space" (ie the diffraction pattern), and vice versa. Between these two spaces, that is, between the two mathematical functions that define them, there is a Fourier transform, but this issue will be discussed in other chapters...
It is necessary to remember that the Fourier transform has been one of the fundamental tools for the development of modern crystallography, and it is worth reading the excellent article published on the legacy of J.-B. J Fourier. But let's go back...