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 more than 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 paper, we focus our attention on discrete-time sparse signals (of length $n$ ). We first show that if the discrete Fourier transform dimension is greater than or equal to $2n$, then almost all signals with 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: identifying the support, i.e., the locations of the nonzero components, of the signal using a combinatorial algorithm; and identifying the signal values in the support using a convex algorithm. We show that TSPR can provably recover most $O(n^{1/2-{\epsilon }})$ -sparse signals (up to a time shift, conjugate flip, and global phase). We also show that, for most $O(n^{1/4-{\epsilon }})$-sparse signals, the recovery is robust in the presence of measurement noise. These recovery guarantees are asymptotic in nature. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR.