Fast and recursive algorithms for magnitude retrieval from DTFT phase at irregular frequencies

We derive two new algorithms for reconstructing a discrete-time 1-D signal from the phase of its discrete-time Fourier transform (DTFT) at irregular frequencies. Previous algorithms for this problem have either required the computation of a matrix nullspace, requiring O(N³) computations, or have been iterative in nature; for the latter, the irregularity of the frequency samples precludes use of the fast Fourier transform. Our first algorithm requires only O(N²) computations (O(N log³ N) asymptotically). In the special case of equally-spaced frequency samples, it is related to a previous algorithm. The second algorithm is recursive-at each recursion a meaningful magnitude retrieval problem is solved. This is useful for updating a solution; it also allows checking of the result at each recursion, avoiding any errors due to computational roundoff error and ill-conditioning of the problem