FOURIER TRANSFORM FOR IRREGULAR FREQUENCY SAMPLING

At any angular frequency $\omega$, the Fourier transform of a discrete, regular function of time is given by the following summation:  

$$F(\omega) = \sum_{n=0}^{N-1} f_n e^{- i \omega n \Delta t}$$ (1)

where N is the number of time samples, $\Delta t$ is the time sampling rate, and f_n is the discrete signal. We notice that equation (1) relates a discrete signal to its continuous Fourier transform. This expression can also be interpreted as a dot product between the discrete time function and a vector of powers of exponentials that depend on the particular frequency $\omega$.Then, for a set of frequencies $\omega_p$, the estimates of the Fourier transform result from a matrix multiplication with the discrete time function.

The implementation of this method is straightforward. The calculation of the matrix of exponentials is coded in parallel, for all frequencies $\omega_p$. A built-in function of Fortran 90 performs an efficient vector-matrix multiplication directly yielding the remapped frequency domain.