Methoden der Kanalcodierung für Compressed Sensing
Final Report Abstract
The two research fields of Channel Coding (CC) and Compressed Sensing (CS) share several commonalities. The maybe most obvious similarity relates to the fundamental underlying problem of CS and syndrome decoding in CC: To obtain sparse solutions to under-determined linear systems is the goal in both research fields. In CC such a sparse vector represents an potential error, whilst in CS, it is often assumed to be some signal representation. However, there is naturally also a significant difference: CC considers usually finite fields, whilst CS operates over continues complex- or real-valued vector spaces. Within this project, the commonalities and differences of the two research fields should be examined in a first step. Afterwards, it should be investigated how methods of the mature field of CC can be used for CS which is a rather young scientific field. In the first project phase, a often described CS-based error correction scheme was evaluated against the background of conventional CC and communication theory. We showed that the considered channel model is far from being realistic and that the implicitly assumed infinite channel capacity could also be exploited by standard digital communication techniques. Additionally, a code concatenation scheme has been proposed which allows the application of CS-based error correction in the case of conventional AWGN channels. The main result of this project is the development of a coherence optimization technique which is based on an algorithm used to determine best spherical codes. The proposed method can be applied to real- and complex-valued vector spaces. It’s fundamental principle is based on the distribution of points altogether with their antipodals on a unit-sphere, where the angular distance between all points and antipodals should be maximized. Due to the generality of the proposed approach, it might find also applications in other research fields. In CS, the method can be applied to solve several relevant optimization problems: For example, it can be used in order to find directly optimized sensing matrices with a close to optimal coherence for arbitrary matrix sizes. Another example is the adaptation of measurement matrices to given dictionaries. It can also be used to provide unequal protection to sparse vector coefficients, which is advantageous for non-uniformly distributed supports. Besides this, a sparsity-aware Simplex algorithm has been proposed which improves on the well known basis pursuit algorithm by utilizing typically avoided degenerated corners in the Simplex algorithm. We investigated deterministic CS systems based on complex Reed–Solomon codes. Thereby, we considered the application of power decoding (virtual interleaving) and proposed a iterative error evaluation algorithm which is especially powerful in noisy environments. Therein, the continuity of the vector space can be used to provide additional reliability information.
Publications
- “Performance of error correction based on compressed sensing,” in Proceedings of 8th International Symposium on Wireless Communication Systems (ISWCS), Aachen, Germany, Nov. 2011, pp. 301–305
H. Zörlein, D. E. Lazich, and M. Bossert
- “Concatenated compressed sensing-based error correcting codes,” in Proceedings of 9th International ITG Conference on Systems, Communication and Coding (SCC), Munich, Germany, Jan. 2013
H. Zörlein, M. Shehata, and M. Bossert
- “Low coherence sensing matrices based on best spherical codes,” in Proceedings of 9th International ITG Conference on Systems, Communication and Coding (SCC), Munich, Germany, Jan. 2013
D. E. Lazich, H. Zörlein, and M. Bossert
- “On the Noise-Resilience of OMP with BASC-Based low coherence sensing matrices,” in Proceedings of 10th International Conference on Sampling Theory and Applications (SampTA), Bremen, Germany, Jul. 2013, pp. 468–471
H. Zörlein, D. E. Lazich, and M. Bossert
- “Deterministic compressed sensing with power decoding for complex reed–solomon codes,” in 10th International ITG Conference on Systems, Communication and Coding (SCC), Hamburg, Germany, Feb. 2015
M. Mohamed, S. Rizkalla, H. Zörlein, and M. Bossert
- “Sparsity aware simplex algorithms for sparse recovery,” in 10th International ITG Conference on Systems, Communication and Coding (SCC), Hamburg, Germany, Feb. 2015
H. Zörlein, T. Leichtle, and M. Bossert
