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We address the problem of recovering a signal (up to global phase) from itsshort-time Fourier transform (STFT) magnitude measurements. This problem arisesin several applications, including optical imaging and speech processing. Inthis paper we suggest three interrelated algorithms. The first algorithmestimates the signal efficiently from noisy measurements by solving a simpleleast-squares (LS) problem. In contrast to previously proposed algorithms, theLS approach has stability guarantees and does not require any prior knowledgeon the sought signal. However, the recovery is guaranteed under relativelystrong restrictions on the STFT window. The second approach is guaranteed torecover a non-vanishing signal efficiently from noise-free measurements, undervery moderate conditions on the STFT window. Finally, the third methodestimates the signal robustly from noisy measurements by solving asemi-definite program (SDP). The proposed SDP algorithm contains an inherenttrade-off between its robustness and the restrictions on the STFT windows thatcan be used.

A Least Squares Approach for Stable Phase Retrieval from Short-Time Fourier Transform Magnitude