In my task I need to perform Inverse Fourier Transform from spectrum that depend on time and frequency arguments simultaneously. E.g., I have a discrete spectrum of some function $S(t, f)$ with $2N$ harmonics counts that's required to be transformed to the signal $U(t)$. So: $$ U(t) = \sum_{k=-N}^NS(t,f_k)e^{2\pi if_kt} $$ Of course, for computation purpose $t$ must be represented as discrete variable (array) and the final signal $U(t)$ also discrete function. $S(t,f_k)$ is a some matrix with corresponding to the time-$n$ and frequency-$k$ values. It should be noted function $S(t,f_k)$ that determine values of this matrix can't be described analytically in the common case.
Using simplest algorithm (straight summation with all complex exponent calculation) it is easy to evaluate numerically on predefined $t$ grid. But computation complexity is very high and if a lot of such calculation required (as for my case) it is very time expensive.
If $S(t,f_k)$ not depend on time argument then I could use a FFT algorithm (even more, for example, fftw3 library). But in the case that was presented above I can't use it because for different times spectrum is different. Also it should be noted that recursive Cooley-Tukey FFT algorithm is not easy to implement because indexes of the time's samples for spectrum function hard to determine at the last recursion levels.
Note, that this situation can be rephrase for forward Fourier Transform so that frequency of signal depend on time (frequency modulated signal), e.g. $cos$: $$ S(f_k)=\int_{-T}^Tcos(2\pi tf(t))dt $$ where $T$ is half of the function period, $f(t)$ - some unknown function that can be specified by mean of Lookup Table or something else.
So, the question is: could be presented expression transformed so that standard FFT algorithm can be applied to it for signal from spectrum calculation? Or may be some algorithm exists to solve it in a convenient (and fastest) way?
P.S. Sorry for my terrible English.