Reconstructing randomly sampled signals by the FFT

The frequency spectrum of a signal that is sampled by a random sampling scheme can be obtained by applying the DFT to the sample sequence. A direct inverse transform on the resulting spectrum, however, will not restore the sample sequence since the orthogonality of the transform kernel is destroyed by the randomization. Apart from orthogonality, an inverse procedure has to deal with other problems, namely, sequence length and noise. In this paper, we propose a procedure to reconstruct from the spectrum a regularly spaced sequence that is equivalent to the original random sequence. Windowing is involved and the FFT can be used for the computation of the inverse.