  We advocate a compressed sensing strategy that consists of multiplying the
signal of interest by a wide bandwidth modulation before projection onto
randomly selected vectors of an orthonormal basis. Firstly, in a digital
setting with random modulation, considering a whole class of sensing bases
including the Fourier basis, we prove that the technique is universal in the
sense that the required number of measurements for accurate recovery is optimal
and independent of the sparsity basis. This universality stems from a drastic
decrease of coherence between the sparsity and the sensing bases, which for a
Fourier sensing basis relates to a spread of the original signal spectrum by
the modulation (hence the name "spread spectrum"). The approach is also
efficient as sensing matrices with fast matrix multiplication algorithms can be
used, in particular in the case of Fourier measurements. Secondly, these
results are confirmed by a numerical analysis of the phase transition of the
l1- minimization problem. Finally, we show that the spread spectrum technique
remains effective in an analog setting with chirp modulation for application to
realistic Fourier imaging. We illustrate these findings in the context of radio
interferometry and magnetic resonance imaging.
