As far as i know the tone is a basic unit of building signals. It is thought that the complex signals is a superposition of such tones. If the signal is composed of periodic repetitive pulses extending on infinite time, then one can analyse this signal by the Fourier series. May be when the signal is limited in space, the amplitude envelope of the sinusoidal waves will be that sinc x function.

In case of single rectangular pulse in time domain which is confined in time T, then it can be composed of an infinite number of sinusoidal waves with a sinx envelope extending in infinite frequency domain. In this case, one no longer has discrete frequencies but one can speak of energy contained in a specified ranges of frequencies. Or one can speak of amplitude samples in the frequency domain. Expressed in other way, there will be smeared frequency spectrum rather than discrete frequencies. This smearing will be less when the pulses widen in time. Which means concentration of the signal in time domain lead to more smearing and spreading in frequency domain and vice verse.

This may be a practical prospective rather than philosophical opinion.

Best wishes

Use fft() and dwt() in Matlab or Scilab or Python

Abdelhalim, thanks for your answer. It is an illuminating one.

I am not sure what the question is actually. The theory by Fourier is clean, and allows to decompose a signal, however complex, on a basis of sinusoidal waves. But this is a mathematical theory, a very helpful one. And from this decomposition, you can always reconstruct it perfectly. But the question of its physical meaning has been addressed at least a century ago and this has led to the time-frequency methods, the wavelets analysis,...etc. There is a physical problem basically. As Jean Ville said in the 40's, if you record a piano music, in which you once play a "A" at a given time, you can decompose your record with Fourier, and will obtain coefficients for all frequencies and among them, one for the "A" frequency. The strange thing is that sines waves are infinite, so superposing the "A" frequency sine waves during the reconstruction should produce a continuous "A" sound. But it actually works and the "A" appears only when you have played it, and this is because of a mathematical trick: in the other times, the "A" is destroyed by interference, and this is very probably very different from the reality. Does it apply to turbulence as well? I believe it, yes. All this stuff is well known I suppose. So what was the gist of your question?

My question is philosophical: are these frequencies real (especially in turbulence)?

Hello Raul,

I'm not sure I understand Patrice's example of the note struck on a piano. Fourier series are usually used to decompose a well-behaved periodic function, i.e., functions that extend to infinity in both directions of the time axis. Striking a key of a piano produces an aperiodic function with a definite attack, sustain, and decay time, i.e., the function is time limited. Such an aperiodic function can be analyzed using the methods of Fourier transforms. The Fourier series is a sum of discrete harmonics (frequencies), while the Fourier transform is usually a function of continuous frequencies. While the Fourier series decomposition only works with periodic time functions, the Fourier transform approach can be employed with aperiodic and periodic time functions. Using the inverse Fourier transform would recreate the original time limited function without the strangeness (a continuous A note) mentioned by Patrice.

As far as the reality of the decomposed frequencies are concerned, I think that is an interesting question. The reason sinusoidal waves are used to decompose periodic and aperiodic time functions is that the homogenous portion of the differential equation (DE) underlying the systems producing the time functions have constant coefficients in all their terms, and so the solutions are unique and linear superpositions of sinusoids. If the coefficients of the homogenous portion of the DE's were not constant, then you would either have a nonlinear DE with a solution that was not necessarily unique nor amenable to linear superposition, or you would have a linear DE whose homogenous solution (as opposed to the particular solution) was something like Bessel functions instead of sinusoids. If the coefficients of the homogeneous part of the DE are functions of the dependent variable, then the DE is nonlinear; if the coefficients are functions of the independent variable, then the DE is linear but with nonsinusoidal solutions (see Henning F. Harmuth; Nonsinusoidal Waves for Radar and Radio Communication; Academic Press; 1981).

In the case of nonlinear DE's, it is not even clear that a frequency decomposition, if it can be done, would even be unique as such DE's do NOT (I forgot the 'not' in my original reply, sorry!) necessarily obey the uniqueness and superposition theorems with respect to their solutions.

Regards,

Tom Cuff

Thanks, Tom. Your answer is illuminating.

To summarize:

1) Any function can be expressed as a linear combination of the elements which constitute an orthogonal basis.

2) The Fourier basis is composed of an infinite number of complex exponential functions.

3) Fourier analysis performs well if the sum of the first elements of the Fourier basis converges to the analyzed function. This occurs when the studied function is periodical and when the sampling rate respects Nyquist's criterion.

4) Fourier analysis becomes important in acoustics and optics because complex exponentials are the eigenfunctions of the wave equation which governs acoustics and optics phenomena. ie harmonic excitation will produce harmonic solution, which can be represented adequately in the Fourier domain.

5) For transient phenomenon, other basis (wavelets for example) can be used to converge faster towards the solution.