Channel Coding Inspired Contributions to Compressed Sensing

This thesis presents three independent contributions to the large and juvenile field of Compressed Sensing (CS). Thereby, a close connection to the mature field of channel coding is established with an interdisciplinary motivation. Both fields are united by their search for a unique sparsest solution to an underdetermined System of Linear Equations (SLE). Within the first contribution, Sparsity Aware Simplex Algorithms (SASAs) are provided which extend the well-known approach of Basis Pursuit (BP). The common concept of BP resembles a convex `1-relaxation to the non-deterministic polynomial-time hard sparse reconstruction problem. In case this relaxation does not lead to the sparsest solution, BP fails inevitably. By extending the famous simplex method for linear optimization, the sparsest solution might be found nevertheless. Thereby, the circumstance is exploited that the desired solution is contained in a degenerated vertex with high probability for systems in general position. All proposed SASAs favor such a degenerated solution against the `1-relaxation and improve thereby the recovery performance, where the variants differ in their trade-off between complexity and potential gain. The second contribution generalizes a minimal distance maximization approach for realvalued spherical codes to vector spaces over complex numbers. Furthermore, an equivalence relation is introduced by the concept of antipodal spherical codes, which ensures that vector pairs of minimal coherence correspond to those of largest minimum distance for these antipodal codes as it is proven within this thesis. Consequently, this relation can be used to extend the aforementioned distance optimization approach to the problem of coherence minimization for the case of antipodal spherical codes. The resulting Best Antipodal Spherical Code (BASC) search approach obtains vector sets which improve significantly on previous numerical results and are often close to the theoretical limit. Since the coherence is an important and limiting factor in many applications, the proposed BASC search approach is not only relevant for CS, where coherence is commonly used as uniqueness assuring property for sensing matrices. Beyond the direct CS application with respect to optimized sensing matrices, the potential use for adapting a measurement matrix to a given dictionary is also investigated. Thereby, the influence of two different coherence-based adaptation strategies is examined. The application of Complex-valued Reed–Solomon (CRS) codes as a deterministic CS scheme is described in details within the third contribution. Previous results in this direction have been extended, whereby particular focus has been put on noise resilience, as this is an often criticized weakness of CRS-based approaches. Within this chapter, power decoding methods have been adapted for three types of error locator algorithms: Peterson’s, Berlekamp–Massey, and extended Euclidean. Together with two error evaluator algorithms, namely Gorenstein–Zierler and Forney’s, the potential application in corresponding deterministic CS schemes has been considered. In order to counter the observed noise sensitivity, an iterative erasure and evaluation algorithm can be used which aims to determine the correct error locations. In contrast to conventional channel codes over finite fields, a low-degree Padéapproximation can be used for complex vector spaces to obtain reliability-like information on the error locations, which is subsequently used by continuity assisted decoding in order to decode even beyond the power decoding radius.