Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

In this paper, we develop a robust uncertainty principle for finite signals in C which states that for nearly all choices T,Ω ⊂ {0, . . . , N − 1} such that |T |+ |Ω| (logN)−1/2 ·N, there is no signal f supported on T whose discrete Fourier transform f̂ is supported on Ω. 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 f̂ is concentrated on Ω. 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 f(s) = ∑ t∈T α1(t)δ(s− t) + ∑