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AbstractIn this work we consider one-dimensional blind deconvolution with prior knowledge of signal autocorrelations in the classical framework of polynomial factorization. In particular this univariate case highly suffers from several non-trivial ambiguities and therefore blind deconvolution is known to be ill-posed in general. However, if additional autocorrelation information is available and the corresponding polynomials are co-prime, blind deconvolution is uniquely solvable up to global phase. Using lifting, the outer product of the unknown vectors is the solution to a (convex) semi-definite program (SDP) demonstrating that -theoretically- recovery is computationally tractable. However, for practical applications efficient algorithms are required which should operate in the original signal space. To this end we also discuss a gradient descent algorithm (Wirtinger flow) for the original non-convex problem. We demonstrate numerically that such an approach has performance comparable to the semidefinite program in the noisy case. Our work is motivated by applications in blind communication scenarios and we will discuss a specific signaling scheme where information is encoded into polynomial roots.
CitationWalk P, Jung P, Hassibi B (2017) Constrained blind deconvolution using Wirtinger flow methods. 2017 International Conference on Sampling Theory and Applications (SampTA). Available: http://dx.doi.org/10.1109/sampta.2017.8024425.
SponsorsWe would like to thank Kishore Jaganathan, Anatoly Khina and Tom Szollmann for helpful discussions. This work was partially supported by the DFG grant JU 2795/3 and WA 3390/1. The work of Babak Hassibi was supported in part by the National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASA’s Jet Propulsion Laboratory through the President and Director’s Fund, by King Abdulaziz University, and by King Abdullah University of Science and Technology.