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Uncertainty principles and ideal atomic decomposition (2001)
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| Venue: | IEEE Transactions on Information Theory |
| Citations: | 580 - 20 self |
BibTeX
@ARTICLE{Donoho01uncertaintyprinciples,
author = {David L. Donoho and Xiaoming Huo},
title = {Uncertainty principles and ideal atomic decomposition},
journal = {IEEE Transactions on Information Theory},
year = {2001}
}
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Abstract
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 1ft = g and sinusoids expf2 iwt=N) = p 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 `1 norm of the coe cients among all decompositions. Here \highly sparse " means that Nt + Nw < p N=2 where Nt is the number of time atoms, Nw is the number of frequency atoms, and N is the length of the discrete-time signal.
Keyphrases
ideal atomic decomposition uncertainty principle discrete-time signal spike sequence time frequency dictionary precise collection sparse superposition nt nw sparse representation combined time frequency dictionary made unique way time atom coe cients convex optimization problem frequency atom
