Spikes, Roots, and Modulations: Phase Retrieval for Finitely-Supported Complex Measures

We study the recovery of a finitely supported distribution, a complex linear combination of Dirac measures, from intensity measurements. The distribution $\mu=\sum_{j=1}^{s}c_{j}\delta_{t_{j}}$ is given by a coefficient vector $c\in\mathbb{C}^s$ and its support $\{t_1, t_2, \dots, t_s\}$ is contained in $[0,\Lambda]$ for some $\Lambda>0$. The intensity measurements evaluate (squared) magnitudes of a set of linear functionals applied to $\mu$, obtained by sampling $\hat \mu$, the Fourier transform of $\mu$, or by evaluating differences between modulated samples. Following a strategy by Alexeev et al., the structure of the linear functionals, and hence of the non-linear magnitude measurement, is encoded with a graph, where the vertices represent point evaluations of $\hat \mu$ at $\{v_1, v_2, \dots, v_n\} \subset [-\Omega,\Omega]$ and each edge represents a (modulated) difference between vertices incident with it. We show that a Ramanujan graph with degree $d \ge 3$ and $n>\frac{6(1 + 6 /\ln(s/\Lambda\Omega)) s}{1-2\sqrt{d-1}/d}$ vertices provides $M=(d+1)n$ magnitudes that are sufficient for identifying the complex measure up to an overall unimodular multiplicative constant. At the cost of including an additional oversampling step and with an additional requirement that $n-1$ is prime, we construct an explicit recovery algorithm that is based on the Prony method.