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 short terms, the proposed ANN approximate approach and the reference algorithm generate BCASCs of equivalent quality in terms of coherence. In other words, Tables and confirm that the reduction of the algorithm complexity has no significant effect on the quality of the obtained BCASCs. In fact, both alternatives often produce codes with equal coherence, closely approaching and eventually meeting the theoretical lower bound. Cases for which our approach meets the theoretical lower bound are @math , @math and @math , @math . Furthermore, in some cases the proposed approach outperforms the reference method, which is a remarkable fact, even when the differences are rather negligible in general. The reader might observe that the coherences obtained for the reference method often do not coincide with those given in @cite_15 . This is due to the fact that we use the variant with approximate discrete summation for calculating the integral over complex rotations, in order to enable a fair comparison to the proposed approach, while the results presented in Tables I and II of @cite_15 were obtained using numerical integration.
@cite_15
2169273233
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.