Compressed sensing with corrupted Fourier measurements

This paper studies a data recovery problem in compressed sensing (CS), given a measurement vector b with corruptions:     0 0 b Ax f   , can we recover   0 x and   0 f via the reweighted 1  minimization: , 1 1 min || || || , . || . x f x f s t Ax f b     ? Where the m n  measurement matrix A is a partial Fourier matrix,   0 x denotes the n dimensional ground true signal vector,   0 f denotes the -dimensional corrupted noise vector, where a positive fraction of entries in the measurement vector b are corrupted by the non-zero entries of   0 f . 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   0 x in these papers. In this paper, we show that   0 x and (0) f can be recovered exactly by the solution of the above reweighted 1  minimization with high probability provided that       0 2 0 log m O x n  I and n is prime, here   0 0 x denotes the cardinality (number of non-zero entries) of   0 x . Except the sparsity, no extra assumption is needed for   0 x .