Subsampling at Information Theoretically Optimal Rates

Details Subsampling at Information Theoretically Optimal Rates Subsampling at Information Theoretically Optimal Rates

2012-02-12

arxiv.org/abs/1202.2525v1

Computer Science

Information Theory

5 pages, 4 figures, submitted to ISIT 2012

Scientific paper

We study the problem of sampling a random signal with sparse support in frequency domain. Shannon famously considered a scheme that \emph{instantaneously} samples the signal at equispaced times. He proved that the signal can be reconstructed as long as the sampling rate exceeds twice the bandwidth (Nyquist rate). Cand\`es, Romberg, Tao introduced a scheme that acquires \emph{instantaneous} samples of the signal at random times. They proved that the signal can be uniquely and efficiently reconstructed, provided the sampling rate exceeds the frequency support of the signal, times logarithmic factors. In this paper we consider a probabilistic model for the signal, and a sampling scheme inspired by the idea of \emph{spatial coupling} in coding theory. Namely, we propose to acquire \emph{non-instantaneous} samples at random times. Mathematically, this is implemented by acquiring a small random subset of Gabor coefficients. We show empirically that this scheme achieves correct reconstruction as soon as the sampling rate exceeds the frequency support of the signal, thus reaching the information theoretic limit.

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