Before covering an important concept in crystallography, the reciprocal lattice, it is worthwhile to consider a Fourier transform and its implications for coordinate systems. This is one of many ways to write a Fourier series:
\[{I(x)}=\sum _{N=-\infty }^{\infty }f_N\exp [2\pi \ iNx/P]\]
In this \(I(x)\) is a function that is periodic in P, meaning that \(I(x) = I(x+P) = I(x+nP)\) where \(n\) is any integer. Or, in words, our periodic function is one that repeats itself at intervals of \(P\). The physical importance of this is that any periodic function can be constructed by adding together exponential (or equivalently sine and cosine) functions, where the lowest order function has the same periodicity as the original periodic function (where N is 1) and all the remainders are harmonics of that, where the periodicity is double, triple,… (thanks to the N term.)
The best example I can offer for the use of the Fourier transform in one dimension has to do with sound. As we know, musical instruments can all play the same notes, where a note has a defined frequency, or equivalently a wavelength, but can sound very different: consider the “pure” sound of a middle-C note from a flute compared to the very “brassy” sound of a middle-C from a saxophone. This note, if sustained so that it sounds the same, is an example of our periodic function. We can build either the flute or saxophone sound by adding together sine waves with the frequency of middle-C (usually 262 Hz) and its harmonics (524 Hz, 786 Hz,…). The harmonics are each an octave above the original note, so we can create the sound of our saxophone by adding together “C” notes with varying amplitudes, as well as some other overtones with related frequencies. (The flute sound is already very close to a pure sine wave, so it needs only very minor harmonic additions.) Thus, we have taken a complex time-varying wave form and expressed it as a mixture of components described by frequency.
The Fourier transform expresses this concept mathematically. Using our example with sound, it takes a signal in units of time, and expresses it as a function in frequency (1/time). This is the magic property of a Fourier transform in that it allows us to go between two sets of coordinates, where one is the inverse of the other. This relationship is described by saying that frequency is the Fourier conjugate of time. This relationship exists not only for one dimensional functions, but also can be expressed in higher dimensions, for example for a three-dimensional (or even six-dimensional) periodic function. We call the domain of the Fourier component functions, which have inverse units from our original function, as operating in “Fourier space”.