Leveraging Diversity and Sparsity in Blind Deconvolution
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1610.06098.pdf
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1.4 MB
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62358762b1a733951f413d3dd0da3e22
Author(s)
Ahmed, Ali
Demanet, Laurent
Date Issued
June 2018
Journal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Ahmed, Ali and Laurent Demanet. “Leveraging Diversity and Sparsity in Blind Deconvolution.” IEEE Transactions on Information Theory (2018) 64, 6: 3975 - 4000 © IEEE
Version
Original manuscript
Abstract
IEEE This paper considers recovering L-dimensional vectors w, and x1, x2,...,xN from their circular convolutions yn = w * x [subscript n]; n = 1, 2, 3,...,N. The vector w is assumed to be S-sparse in a known basis that is spread out in the Fourier domain, and each input x[subscript n] is a member of a known K-dimensional random subspace. We prove that whenever K + S log[superscript 2]S ≲ L/log[superscript 4](LN), the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey N ≳ log[superscript 2](LN). By "diverse inputs", we mean that the x[subscript n]'s belong to different, generic subspaces. To our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which w belongs is not fixed, but needs to be determined. We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients, and the delays in the channel impulse response w are unknown. The encoder codes the K-dimensional message vectors randomly and then transmits coded messages x[subscript n]'s over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than (K + Slog[superscript 2]S) log[superscript 4](LN), and the number of messages is roughly at least log2(LN).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/TIT.2017.2788444
