The Fourier transformation is one of the most frequently used techniques in mathematical physics, which allows for instance to transform any time dependent function f(t) into its frequency spectrum

It is only the inclusion of the phase factor which in general allows the inverse Fourier transform

For instance, if one has a Gaussian frequency spectrum centered at frequency ω_{0} i.e.
_{0} and a damping constant γ (assumed here independent of frequency) subject to Doppler broadening with a width Δω: if the individual wave trains are offset in time by a (random) amount t_{0}(ω), then their superposition yields the time signal (leaving out the normalization constants here)

Now, since t_{0}(ω) is a (random) function of ω, it is obvious that t-t_{0}(ω) can not simply be replaced by a variable t' such that Eq.(7) could be interpreted as the Fourier transform of exp[-(ω-ω_{0})/ Δω)^{2}] in terms of t'. On the other hand, we know that for those wave trains that overlap significantly at all, |t_{0}(ω)| has on average a value of the order of 1/γ. So if we consider only times t<<1/γ, we can approximately replace iω(t-t_{0}(ω)) by -iωt_{0}(ω), and thus we have

The integral over the individual emissions is now not only independent of t, but, since t_{0}(ω) is a random function, independent of the Doppler broadening Δω altogether. This is because the 'phase factor' exp[-iωt_{0}(ω)] is now not only not sinusoidal anymore (which it would only be for t_{0} independent of ω), but merely provides a random factor for each emitted wave train with a value between -1 and 1 (for the real component). Since the Gaussian Doppler broadening distribution is normalized (if one includes the here neglected normalization factor), this means that if the distribution is made up of a large number N of overlapping wavetrains, the integral will, according to the laws of statistics, be of the order of √N (the factor exp[γt_{0}(ω)] will on average be of the order of 1), i.e.

On the other hand, the coherency can of course be arbitrarily decreased (i.e. the bandwidth increased) by 'chopping' the signal into smaller segments (e.g. by means of an electro-optical shutter).

(1) F(ω) = _{-∞}∫^{∞}dt f(t)^{.}e^{-iωt} .

(2) F(ω) = |F(ω)|^{.}e^{iφ(ω)} ,

It is only the inclusion of the phase factor which in general allows the inverse Fourier transform

(3) f(t) = 1/2π ^{.}_{-∞}∫^{∞}dω F(ω)^{.}e^{iωt} .

(4) Δω^{.}Δt = 1/2 .

For instance, if one has a Gaussian frequency spectrum centered at frequency ω

(5) |F(ω)| ~ e^{-((ω-ω0)/ Δω)2} .

(6) f(t) ~ e^{iω0t}^{.}e^{-(t/Δt)2} .

(7) f(t) ~ _{-∞}∫^{∞}dω e^{-((ω-ω0)/ Δω)2}^{.}e^{-γ.(t-t0(ω))}^{.}e^{iω(t-t0(ω))} ,

Now, since t

(8) <f(t)> ~ e^{-γt}^{.}_{-∞}∫^{∞}dω e^{-((ω-ω0)/ Δω)2}^{}e^{γt0(ω)}^{.}e^{-iωt0(ω)} (t<<1/γ) .

The integral over the individual emissions is now not only independent of t, but, since t

(9) <f(t)> ~ e^{-γt}^{.}√N .

(10) f(t) ~ e^{-((ω'-ω0)/ Δω)2}^{.}e^{-γ.(t-t0(ω'))}^{.}e^{iω(t-t0(ω'))}

On the other hand, the coherency can of course be arbitrarily decreased (i.e. the bandwidth increased) by 'chopping' the signal into smaller segments (e.g. by means of an electro-optical shutter).

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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See also my sister site https://www.plasmaphysics.org.uk