Inverting spectrogram measurements via aliased Wigner distribution deconvolution and angular synchronization

摘要

We propose a two-step approach for reconstructing a signal x ∈ C~d from subsampled discrete short-time Fourier transform magnitude (spectogram) measurements: first, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix xx*. Secondly, we use angular synchronization to solve for x (and then for x by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems; one which guarantees the recovery of discrete, bandlimited signals x ∈ C~d from fewer than d short-time Fourier transform magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.