Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms

The problem of signal recovery from its Fourier transform magnitude is of paramount importance in various fields of engineering and has been around for over 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this work, we focus our attention on discrete-time sparse signals (of length $n$). We first show that, if the DFT dimension is greater than or equal to $2n$, almost all signals with {\em aperiodic} support can be uniquely identified by their Fourier transform magnitude (up to time-shift, conjugate-flip and global phase). Then, we develop an efficient Two-stage Sparse Phase Retrieval algorithm (TSPR), which involves: (i) identifying the support, i.e., the locations of the non-zero components, of the signal using a combinatorial algorithm (ii) identifying the signal values in the support using a convex algorithm. We show that TSPR can {\em provably} recover most $O(n^{{1/2-\eps})$-sparse} signals (up to a time-shift, conjugate-flip and global phase). We also show that, for most $O(n^{{1/4-\eps})$-sparse} signals, the recovery is {\em robust} in the presence of measurement noise. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR.