  We study the problem of sampling a random signal with sparse support in
frequency domain. Shannon famously considered a scheme that 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 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 spatial coupling in coding theory.
Namely, we propose to acquire 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.
