A hybrid Fourier-Prony method

The discrete Fourier transform is one of the top ten algorithms of the 20-th century, frequently used in the analysis of signals. It is known to be very stable and of low computational complexity. When the bandwidth is large or the spectrum is sparse, sampling at the Shannon-Nyquist rate may however be prohibitive. We propose a new hybrid technique that exploits the spectral sparsity. Several shifted and decimated but uniformly sampled data sets are collected. The aliased discrete Fourier spectra are corrected for the aliasing effect by a combination with Prony’s method. In our approach both the decimation factor and the time shift can be large and thus violate the Shannon-Nyquist theorem. The resulting technique delivers the same accuracy as the non-decimated Fourier transform, uses fewer samples and achieves a smaller computational cost. Also, the method is suitable for parallellization.