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.
In this section we provide a general comparison between the ANN-BCASC approximation and other approaches for generating close-to-optimal complex codes. In order to provide a complete cumulative comparison, the best option for such comparison is to extend the tables in @cite_15 , which are in turn an extension of Table II of @cite_39 and Table II of @cite_77 , respectively. These tables provide the values of coherence obtained from complex SCs generated using different methods and are named Table and Table , homonimously to the corresponding tables in @cite_15 . For both the BCASC and the ANN-BCASC methods we adopt the best result of ten independent runs. We also measure the time that each algorithm needed to generate the codes. For the other algorithms, we preserve the values in the original tables of @cite_15 . We use the reference BCASC algorithm with approximate summation with @math summands, in order to have a fair reference for the ANN-BCASC algorithm, for which we adopt the same @math and a constant @math for all cases considered. As in @cite_15 , for both methods @math and @math were used.
@cite_77 @cite_15 @cite_39
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Grassmannian quantization codebooks play a central role in a number of limited feedback schemes for single and multi-user multiple-input multiple-output (MIMO) communication systems. In practice, it is often desirable that these codebooks possess additional properties that facilitate their implementation, beyond the provision of good quantization performance. Although some good codebooks exist, their design tends to be a rather intricate task. The goal of this paper is to suggest a flexible approach to the design of Grassmannian codebooks based on sequential smooth optimization on the Grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties. As one example, the proposed approach is used to design rank-2 codebooks that have a nested structure and elements from a phase-shift keying (PSK) alphabet. In some numerical comparisons, codebooks designed using the proposed methods have larger minimum distances than some existing codebooks, and provide tangible performance gains when applied to a simple MIMO downlink scenario with zero-forcing beamforming, per-user unitary beamforming and rate control (PU 2RC), and block diagonalization signalling. Furthermore, the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance. 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. 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.