Difference set sampling for Source Localization and Beyond
High Resolution source localization is an important problem arising in applications across multiple disciplines. The goal is to identify the location of point sources from plane waves captured at an array of sensors in the form of a space-time series. By associating a qth (q ≥ 2) order difference set corresponding to the physical sensor locations, I will demonstrate its role in source localization from qth order statistics of the spatiotemporal data. In particular, it is possible to localize O(Mq) sources with only M sensors, by using new geometries for spatial samplers. The difference set also plays important role in determining the robustness of spatial samplers against perturbation or calibration errors. In particular, the redundancies present in the difference set can be exploited to ensure identifiability of source parameters in presence of such errors. Finally, the role of 2nd order difference sets in complex phase retrieval will be demonstrated, inspiring the design of a new class of non uniform Fourier based sampler, that can provably recover a complex signal (upto a global phase ambiguity) from its amplitude measurements with near-minimal number of samples. An interesting connection with the 4N-4 conjecture will also be established, which hypothesis 4N-4 to be the minimum number of generic measurements necessary to ensure injectivity in N dimensions for complex phase retrieval.