Uncertainty principles and ideal atomic decomposition

Suppose a discrete-time signal S(t), 0⩽t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1_{t=τ} and sinusoids exp{2πiwt/N}/√N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l¹ norm of the coefficients among all decompositions. Here “highly sparse” means that N_t+N_w<√N/2 where N_t is the number of time atoms, N_w is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l¹ uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l¹ norm atomic decomposition. The condition “sufficiently sparse” becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2^j/2