What do the frequencies in the a Fourier transform of a non-periodic signal mean physically?
Are there another definition of frequency that doesn't include the FT?
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The physical siginificance of the Fourier transform for nonperiodic (finite energy) signals is not something clear from any point of view.
The limiting case of a Fourier transform of a periodic signal, as the period goes to infinity, consists of line components (dirac impulses at a countable set of discrete frequencies) which becomes more and more crowded, eventually approaching a continuum of frequencies, at which are no more dirac impulses.
This infinitesimal frequency component at any $\omega$, has no physical significance, unlike the dirac impulse at $\omega$, which has a direct implication of a sinusoidal signal component, extending from minus infinity to infinity (in time for example).
Non-periodic signals are also known as transients, or non-stationary signals, and for their analysis, Wavelet transform family can provide a better insight than the Fourier family.
Nevertheless, the Fourier family of transforms can still be used to analyse periodic as well as nonperiodic signals with great success.