Channel Coding Inspired Contributions to Compressed Sensing
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Abstract
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 second contribution generalizes a minimal distance maximization approach for real-valued 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 application of Complex-valued Reed–Solomon (CRS) codes as a deterministic CS scheme is described in details within the third contribution. Within this chapter, power decoding methods have been applied. In contrast to conventional channel codes over finite fields, a low-degree Pade-approximation 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.
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Fakultät für Ingenieurwissenschaften und Informatik
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DFG Project uulm
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Keywords
Channel coding, Coherence optimization, Complex Reed-Solomon Codes, Compressed sensing, Simplex Algorithm, Codierungstheorie, Kanalcodierung, Komprimierte Abtastung, RS-Code, Simplexverfahren, Coding theory, Reed-Solomon codes, DDC 620 / Engineering & allied operations