In practice, the signal is usually time-varying,like s(t)=Aexp(iw(t)), where A is the amplitude and w(t) is the instantaneous phase. To calculate the Fourier transform of the signal s(t), we can expand the phase w(t) by Tylaor expansion, like w(t)=a0+a1*t+a2*t2+a3*t3...... Then, how to obtain the Fourier transform of such a time-varying signal with polynomial phase?
This is really difficult. In order to be able to do this, you may study the so-called method of "stationary phase" and as a "simple" case of this how to derive the Fourier transform of a chirp signal.
More:
https://en.wikipedia.org/wiki/Stationary_phase_approximation
https://en.wikipedia.org/wiki/Chirp_spectrum
Linear chirps usually have a start time and a stop time (phase has compact support). However, in Serge Lang's book "SL2(R)" you will find the derivation of a Fourier transform of an infinitely extended chirp which is something that cannot exist in reality. It is at least far beyond the powers of our imagination.
Dear Gang You:
since the function s(t)=Aexp(iw(t)), where w(t) is a polinomial belongs to the space S' of
tempered distributions, the Fourier transform of such a time-varying signal should be
considered in the sense of L. Schwartz. E.g., the Fourier transform of Aexp(ict) is A\delta(x+c) (up to a normalization constant ).
Best,
Adolf.
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