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 im- proved sparsity thresholds for recovery of signals that are sparse in general dictionaries. Specifically, our results improve on the well-known (1 + 1/d)/2-threshold for dictionaries with coherence d 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.

We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and recovery of signals that are impaired by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a novel uncertainty relation for pairs of general dictionaries and we provide corresponding practicable recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge about the signal and noise support sets.

We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and the recovery of signals that are corrupted by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a recently developed uncertainty relation and provide corresponding recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge on the support sets of signal and noise.

As shown by Tropp, 2008, for the concatenation of two orthonormal bases (ONBs), breaking the square-root bottleneck in compressed sensing does not require randomization over all the positions of the nonzero entries of the sparse coefficient vector. Rather the positions corresponding to one of the two ONBs can be chosen arbitrarily. The two-ONB structure is, however, restrictive and does not reveal the property that is responsible for allowing to break the bottleneck with reduced randomness. For general dictionaries we show that if a sub-dictionary with small enough coherence and large enough cardinality can be isolated, the bottleneck can be broken under the same probabilistic model on the sparse coefficient vector as in the two-ONB case.

We analyze fading relay networks, where a single-antenna source-destination terminal pair communicates through a set of half-duplex single-antenna relays using a two-hop protocol with linear processing at the relay level. A family of relaying schemes is presented which achieves the entire optimal diversity-multiplexing (DM) tradeoff curve. As a byproduct of our analysis, it follows that delay diversity and phase-rolling at the relay level are optimal with respect to the entire DM-tradeoff curve, provided the delays and the modulation frequencies, respectively, are chosen appropriately.