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.
The construction method first introduced in @cite_81 and further developed and analyzed in @cite_68 exploits the fact that unitary transforms representing any point on the Grassmann manifold can be represented by the exponential of any element of the tangent space at the identity point. For clarity, in the rank one case we deal with in this paper this would mean an exponential of an @math -dimensional vector with only @math nonzero components. This is due to the fact that we pack subspaces of dimension one (lines) on the Grassmannian, which yields a tangent space of dimensionality @math . The core idea of the method is to design a coherent codebook in the tangent space, which yields an optimal non-coherent one on the Grassmannian. Further work exploiting the exponential parametrization of the Grassmannian can be found in @cite_36 , where different rotated lattices are used as initial codes in the tangent space.
@cite_36 @cite_68 @cite_81
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Geometric methods for construction of codes in the Grassmann manifolds are presented. The methods follow the geometric approach to space-time coding for the non-coherent MIMO channel where the code design is interpreted as a packing problem on Grassmann manifolds. The differential structure of the Grassmann manifold provides parametrization with the tangent space at the identity element. Grassmann codes for the non-coherent channel are constructed by mapping suitable subsets of lattices from the tangent space to the Grassmann manifold via the exponential map. As examples, constructions from the rotated Gosset, Barnes-Wall and Leech lattice are presented. Due to the specifics of the mapping, some of the structure is preserved after the mapping to the manifold. The method is further improved by modifying the mapping from the tangent space to the manifold. Ideas for other constructions of Grassmann codes are also presented and discussed. We construct a new family of space-time codes suited for multiple-antenna non-coherent communications over Rayleigh flat fading channels. These codes use all the complex degrees of freedom of the system, i.e. (M times(1-M T)) symbols per channel use, where T is the code length and M is the number of transmit antennas. Their codewords belong to the Grassmann manifold GTldrM (C), the set of the M-dimensional vector subspaces of CT. Our codes are built from space-time codes for coherent systems via a non linear map (parameterization) called the exponential map. The relationships between some properties of the non-coherent space-time codes and their corresponding coherent codes are investigated. We also propose a simplified decoding for these classes of unitary space-time codes. We analyse the performance of these codes in terms of complexity and error probability and we compare them with some current best propositions, in particular training-based codes. The performance of our codes is substantially equivalent or slightly better than the one of training-based codes, under GLRT (Generalized Likelihood Ratio Test) decoding. When a simplified decoder is applied at the receiver, the error probability curves of the two propositions seem to be basically equivalent in the SIMO case with small block size, high spectral efficiency and high number of receive antennas, and in the 2times2 MIMO case with low spectral efficiency. In the other cases the training codes with simplified decoding seem to outperform our proposition. A family of space-time codes suited for noncoherent multi-input multi-output (MIMO) systems is presented. These codes use all the complex degrees of freedom of the system, i.e. M spl times (1-(M T)) symbols per channel use. They are constructed as codes on the Grassmann manifold G sub T,M ( spl Copf ) where T is the temporal codelength and M is the number of transmit antennas.