  This paper studies a data recovery problem in compressed sensing (CS), given
a measurement vector b with corruptions: b=Ax0+f0, can we recover x0 and f0 via
the reweighted l1 minimization: minimize |x| + lambda*|f| subject to Ax+f=b?
Here the m by n measurement matrix A is a partial Fourier matrix, x0 denotes
the n dimensional ground true signal vector, f0 denotes the m-dimensional
corrupted noise vector, it is assumed that a positive fraction of entries in
the measurement vector b are corrupted by the non-zero entries of f0. This
problem had been studied in literatures [1-3], unfortunately, certain random
assumptions (which are often hard to meet in practice) are required for the
signal x0 in these papers. In this paper, we show that x0 and f0 can be
recovered exactly by the solution of the above reweighted l1 minimization with
high probability provided that m>O(card(x0)log(n)log(n)) and n is prime, here
card(x0) denotes the cardinality (number of non-zero entries) of x0. Except the
sparsity, no extra assumption is needed for x0.
