Compressive Sensing (CS) theory states that real-world signals can often be recovered from much fewer measurements than those suggested by the Shannon sampling theorem. Nevertheless, recoverability does not only depend on the signal, but also on the measurement scheme. The measurement matrix should behave as close as possible to an isometry for the signals of interest. Therefore the search for optimal CS measurement matrices of size @math translates into the search for a set of @math @math -dimensional vectors with minimal coherence. Best Complex Antipodal Spherical Codes (BCASCs) are known to be optimal in terms of coherence. An iterative algorithm for BCASC generation has been recently proposed that tightly approaches the theoretical lower bound on coherence. Unfortunately, the complexity of each iteration depends quadratically on @math and @math . In this work we propose a modification of the algorithm that allows reducing the quadratic complexity to linear on both @math and @math . Numerical evaluation showed that the proposed approach does not worsen the coherence of the resulting BCASCs. On the contrary, an improvement was observed for large @math . The reduction of the computational complexity paves the way for using the BCASCs as CS measurement matrices in problems with large @math . We evaluate the CS performance of the BCASCs for recovering sparse signals. The BCASCs are shown to outperform both complex random matrices and Fourier ensembles as CS measurement matrices, both in terms of coherence and sparse recovery performance, especially for low @math , which is the case of interest in CS.
[Coherence comparison of complex codes II] Comparison of the coherence of close-to-optimal complex codes obtained via different numerical approaches. Based on Table II of @cite_15 .
@cite_15
2169273233 2103374230 2121743251 2155418912
Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases. We introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel E sub b N sub 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece (1996). Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. Consider a codebook containing N unit-norm complex vectors in a K-dimensional space. In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I sub max ) is often desirable. Relying on tools from combinatorial number theory, we construct analytically optimal codebooks meeting, in certain cases, the Welch lower bound. When analytical constructions are not available, we develop an efficient numerical search method based on a generalized Lloyd algorithm, which leads to considerable improvement on the achieved I sub max over existing alternatives. We also derive a composite lower bound on the minimum achievable I sub max that is effective for any codebook size N. Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n,k,d] linear code having all-symbol (r,δ)-locality, denoted as (r,δ)_a, is considered optimal if it has the actual highest minimum distance of any code of the given parameters n,k,r and δ. A minimum distance bound is given in . The existing results on the existence and the construction of optimal (r, δ)_a linear codes are limited to only two small regions within this special case, namely, i) m=0 and ii) m≥ (v+δ-1)>(δ-1) and δ=2, where m=n mod(r+δ-1) and v=k mod r. This paper investigates the properties and existence conditions for optimal (r,δ)_a linear codes with general r and δ. First, a structure theorem is derived for general optimal (r,δ)_a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no (r,δ)_a linear code can achieve the minimum distance bound in . For four other cases the optimal (r,δ)_a codes are shown to exist over a field of size q≥( n k-1 ), deterministic constructions are proposed. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on optimal (r,δ)_a linear codes, leaving only two cases in which the construction of optimal codes are not yet known.