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Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets
We present an uncertainty relation for the representation of signals in two
different general (possibly redundant or incomplete) signal sets. This
uncertainty relation is relevant for the analysis of signals containing two
distinct features each of which can be described sparsely in a suitable general
signal set. Furthermore, the new uncertainty relation is shown to lead to
improved sparsity thresholds for recovery of signals that are sparse in general
dictionaries. Specifically, our results improve on the well-known
-threshold for dictionaries with coherence by up to a factor of
two. Furthermore, we provide probabilistic recovery guarantees for pairs of
general dictionaries that also allow us to understand which parts of a general
dictionary one needs to randomize over to "weed out" the sparsity patterns that
prohibit breaking the square-root bottleneck.Comment: submitted to IEEE Trans. Inf. Theor
A survey of uncertainty principles and some signal processing applications
The goal of this paper is to review the main trends in the domain of
uncertainty principles and localization, emphasize their mutual connections and
investigate practical consequences. The discussion is strongly oriented
towards, and motivated by signal processing problems, from which significant
advances have been made recently. Relations with sparse approximation and
coding problems are emphasized
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