Statement and applications of Fourier's theorem

• Fourier's theorem states that

• In the section harmonic waves, which we invite you to reread before tackling this demanding page, we described a particular type of oscillation called harmonic, described by a particular periodic function called the sine.
• The page on harmonic motion shows that many of the oscillations found in nature are correctly described by this periodic function.
• On this page you will learn that the harmonic oscillation described by the sinusoidal function is even a sort of fundamental periodic oscillation with which to "build", in a sense to be made precise, all kinds of oscillation, periodic and not. This statement is based on a mathematical theorem, well known (to specialists), due to the French mathematical physicist Fourier.
• Building any complex oscillation from the superposition of simple harmonic oscillations is a procedure called synthesis, which is widely used in producing electronic devices able to reproduce the timbre of the various musical instruments or to produce completely artificial sounds (think of the sounds, unheard of and impossible to produce with ordinary musical instruments, of some electronic music).
• The possibility of "decomposing" the complex oscillation into its harmonic oscillations is, so to speak, the inverse procedure of synthesis, and is called spectral analysis or Fourier analysis. This analysis, far from being a mere mathematical curiosity, provides the theoretical basis for countless applications. It underlies
  • the possibility of understanding how the peculiar timbre of each musical instrument arises;
  • the design of speech recognition and speech synthesis systems;
  • the possibility of finding the resonance frequencies of oscillating systems;
  • most spectroscopic techniques in physics: the word spectrum itself refers mostly to the Fourier analysis of a signal. For example, optical spectroscopy is used in the physico-chemical analysis of substances or, in astronomy, as a tool for investigating space and celestial bodies. Spectrum analysis can also be carried out on invisible electromagnetic waves, such as those in the ultraviolet and X-ray bands or, at the other end of the spectrum, radio waves.

Spectrum analysers

Spectral analysis, that is, the determination of the coefficients ${\displaystyle a_n}$ and ${\displaystyle b_n}$, can also be carried out for periodic oscillations whose analytic expression ${\displaystyle y(t)}$ is not known. Such oscillations can be, for example,

• the oscillations of any mechanical system, such as a string or a membrane of a musical instrument, or a bridge or a building subjected to external forces;
• the electrical oscillations of a circuit hit by a complex electromagnetic signal, etc.

In these cases spectral analysis is carried out "empirically", by making the wave in question interact with physical systems able to activate mechanisms that can select and discriminate, separating them, the various spectral components (that is, the various harmonics) of which the wave is made up. These mechanisms can be of various kinds:

• in the phenomena of refraction and dispersion, the selective mechanism is the dependence of the propagation speed of the various harmonics on the frequency of the harmonics themselves. In the very famous prism experiment, Newton split "white" sunlight into a bundle of coloured rays reproducing the colours of the rainbow. In the case of a complex electromagnetic wave, the harmonic components are, not by chance, called monochromatic;
• in the phenomenon of resonance, the selective mechanism is provided by the property of oscillating systems of absorbing energy from the different harmonics more or less efficiently.
  • In the case of sound waves, Helmholtz used this principle to build a set of resonators to analyse the complex sound generated by musical instruments. The monochromatic equivalent of sound is a sound wave made up of a single harmonic, like, to a first approximation, that of the tuning fork. Nowadays there are electronic systems in which the resonance frequency is set by changing physical parameters over which one has better control than over Helmholtz's "mechanical" resonators. For this reason spectral analysis is now carried out by means of oscillating circuits, which are much more reliable, selective and precise than Helmholtz resonators. Our ear too seems to be a very refined spectrum analyser: it discriminates the various harmonics by making different parts of the basilar membrane of the cochlea vibrate, depending on the frequency.
  • In the case of electromagnetic waves, oscillating circuits find applications in receivers (radio, TV, etc.). Indeed, it is clear that when an oscillating circuit is hit by a complex oscillation, that is, not by a single sinusoidal oscillation of frequency f but by the superposition of several oscillations, the effect of resonance is to select the frequencies close to ${\displaystyle f_0}$, attenuating all the others.

Representation of periodic waves in the frequency domain

Fourier's theorem gives us a formidable tool for a concise representation of periodic functions: instead of drawing the graph of the periodic function, it is more effective to represent its spectral content directly by means of spectrograms, that is, by a system of Cartesian axes with the frequencies of the harmonics (all multiples of the frequency of the fundamental) on the horizontal axis and the amplitude of each harmonic on the vertical axis.

The figure shows the graph and the spectral content of the function with equation:

${\displaystyle y(t)=3\cdot \sin (400\pi t)+2\cdot \sin (800\pi t)+\sin (1200\pi t)}$

made up of the superposition (sum) of three harmonic partials with amplitudes (in cm) 3, 2, 1 and frequencies 200, 400, 600 Hz respectively. The graph was drawn over one periodicity interval of the function y(t), which is T = 5 ms, that is, a frequency of 200 Hz (which is that of the fundamental).

Graph in the time domain	Graph in the frequency domain (spectrum)

Below are the graph of the function and the corresponding spectrogram

${\displaystyle y(t)=3\cdot \sin (400\pi t)+2\cdot \sin (800\pi t)+1\cdot \sin (1000\pi t)}$

made up of the superposition (sum) of three harmonic partials with amplitudes (in cm) 3, 2, 1 and frequencies 200, 400, 500 Hz respectively. (This waveform could be obtained from a periodic function with a frequency of 100 Hz in which the first and the third harmonics and all the harmonics above the fifth have been suppressed)

Graph in the time domain	Graph in the frequency domain (spectrum)

In this case too, the graph was drawn over one periodicity interval of the function y(t), which this time is T = 10 ms, that is, a frequency of 100 Hz (which is that of the suppressed fundamental). The general rule is that the frequency of the resulting signal is the greatest common divisor (possibly rational) of the frequencies of the various harmonics. This fact is of great importance in the perception of pitch of a sound, in the phenomenon of virtual pitch.

Some remarks

• It is precisely thanks to Fourier's theorem (which guarantees that the frequencies of the spectrum are all multiples of the fundamental) that the spectral content of the periodic wave appears, in the frequency domain, as a series of vertical lines. The analysis of the spectrograms of periodic functions (as are, to a good approximation, those generated by musical instruments) allows us to grasp many features of the sound produced, such as pitch, loudness and timbre. Below are the waveform, the spectrogram and (why not?) the audio of a real sound (a B♭ at 466 Hz) produced by a trumpet

waveform	spectrogram	Audio
		TR_Sib.mp3 Trumpet: B♭ at 466Hz;

If you want to learn how to read the many pieces of information contained in the graphs above, visit the page How do you read a spectrogram?

• This representation is also useful in the case of non-periodic functions: as explained on the page on uncertainty, a non-periodic function can be "reconstructed" from sine waves of any frequency. In this case the graph of the spectral content, instead of consisting of sharp vertical peaks, will be a continuous distribution, which nevertheless makes it possible to see the frequency ranges that contribute most to forming the non-periodic function.

• Being able to determine the spectral content of non-periodic waveforms too is fundamental, since real waves (such as sound) are not periodic and evolve in time, together with their spectral content. For this reason it is often useful to have a representation of a set of spectrograms, taken at successive times, to analyse the evolution of the spectral content of the wave. Because of its great usefulness in the analysis of sounds, this method is called a sonogram.

Knowing, through sonograms, how the spectral content of a sound evolves in time is very important in determining the timbre of the sound itself.

Check your understanding

Suggested uses of the app: see the paths

To see Fourier's theorem "at work" you can visit the corresponding app. The app is very intuitive and easy to use: on the page Fourier's theorem at work you will find a series of suggestions for using it.

Links and further reading

• If you want to know the formulas for obtaining the coefficients ${\displaystyle a_n}$ and ${\displaystyle b_n}$, visit the page Fourier series expansions; you will also find a "library" of series expansions of particular waveforms (triangle wave, square wave, "sawtooth" waves)
• If you want to see how complex an oscillation can be, visit the sections wave synthesis and normal modes, where you will see the incredible ways in which a string fixed at its ends can vibrate. Fourier's theorem breaks down the complex oscillation into a superposition of simpler oscillations.
• If you want to understand how the timbre of the sounds produced by the various musical instruments arises and evolves, visit the page on perception of timbre