  We develop a robust uncertainty principle for finite signals in C^N which
states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log
N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier
transform is supported on W. In fact, we can make the above uncertainty
principle quantitative in the sense that if f is supported on T, then only a
small percentage of the energy (less than half, say) of its Fourier transform
is concentrated on W.
  As an application of this robust uncertainty principle (QRUP), we consider
the problem of decomposing a signal into a sparse superposition of spikes and
complex sinusoids. We show that if a generic signal f has a decomposition using
spike and frequency locations in T and W respectively, and obeying |T| + |W| <=
C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition
(all other decompositions have more non-zero terms). In addition, if |T| + |W|
<= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving
a convex optimization problem.
