Methoden der Kanalcodierung zur zuverlässigen Datenübertragung und Kanalschätzung bei OFDM/DMT und langsam veränderlichen Kanälen

• Applicant: Professor Dr.-Ing. Martin Bossert
• Subject Area: Electronic Semiconductors, Components and Circuits, Integrated Systems, Sensor Technology, Theoretical Electrical Engineering
• Term: from 2004 to 2009
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5425264

• Final Report Year: 2008

Final Report Abstract

In this project we consider methods of channel coding for wireline communications using orthogonalfrequency-division-multiplexing (OFDM). We have worked on four directions: polyalphabetic codes, interleaved Reed-Solomon codes, trellis coded modulation with signal shaping, and product convolutional codes. Upper and lower bounds on the cardinality of a polyalphabetic code with a given Hamming distance are obtained. Some constructions of polyalphabetic codes are suggested based on known codes. Encoding and decoding of the polyalphabetic codes, obtained in this way, can be done using encoding and decoding algorithms for the mother code. Polyalphabetic codes constructed with Reed-Solomon (RS) mother code, reach the upper Singleton type bound. Some properties of group polyalphabetic codes are investigated. Within the framework of the collaboration with the University of Mannheim, we exchanged our results and software. Simulations results for band-limited linear Gaussian channel with additional frequency selective noise show that polyalphabetic codes (obtained from RS mother code) slightly outperform RS codes (gains of 0.5 dB and more were observed). Our work is devoted to distance profiles of arbitrary codes (polyalphabetic or usual) with unequal error correcting capabilities. We show that for many codes, interesting for practical application, these distance profiles can be obtained using well known MaxLog a posteriori probability decoding algorithms. In we consider trellis-coded QAM modulation (TCM) with signal shaping. We proposed a simple metric modification for the Viterbi decoder of TCM, to take into account the influence of shaping. This allows the increment of the coding gain (e.g. by 0.3 dB for QAM-16) without increasing the decoding complexity. Gray codes of dimension one, two, and even multidimensional Gray codes are frequently used in TCM with (or without) OFDM. We show that every multidimensional Gray code can be uniquely represented as the direct product of one-dimensional Gray codes. Non-existence of Gray labelings for some signal constellations is also proved. Interleaved Reed-Solomon (IRS) codes are interesting for OFDM and some other applications. We show that IRS codes allow the correction of errors beyond half the minimum code distance if the errors are not distributed independently in the received signal but occur in bursts. This is the case when IRS codes are used in concatenated codes. We present such concatenated codes and demonstrate the gain that can be obtained by IRS decoding in comparison to independently decoding of the underlying Reed-Solomon codes. Decoding of IRS codes is based on the problem of finding the shortest linear shift-register capable of generating a number of finite length sequences over some field. This problem was solved. A new approach for decoding a single low-rate Reed-Solomon code beyond half the minimum distance is considered and analyzed. Unlike the Sudan algorithm published in 1997, this new approach is based on multi-sequence shift-register synthesis, which makes it easy to understand and simple to implement. Moreover, the error correcting radius coincides with the error correcting radius of the original Sudan algorithm. Combining known codes is a powerful method to obtain new codes with several advantageous features. For block codes many combining methods are suggested and analyzed. For convolutional codes only a few combining methods have been suggested. We give a simple algebraic definition and investigate the direct product of convolutional codes. Product codes alow product decoding in addition to serial decoding. Simulations show that product decoding converges faster. In our cases 3 iterations of product decoding were enough to get the same performance as serial decoding after more than 10 iterations. Actually, the proposed idea to combine convolutional codes is more general. It is well known that convolutional codes can be considered as block codes over a field of rational functions. Being a block code, every convolutional code has "block" distance. The free distance of a convolutional code is lower bounded by the block distance. With this approach, every method of designing or combining block codes immediately gives a method to design or to combine convolutional codes. We investigate the properties of block distance and demonstrate the proposed idea for Reed-Solomon codes, for the direct product codes and for bipartite graph codes. In addition to the main stream of the project, some interesting preliminary results were obtained in the following two directions: "Combining of convolutional codes", and "Interleaved Reed-Solomon codes". We believe that these areas are fruitful, and research in these directions should be continued not only in the frame of the current project. At the moment the topic "Interleaved Reed-Solomon codes" is investigated under another DFG project.

Publications

• "Decoding of Trellis Coded Modulation with Shaping". 2004 Intern. Symposium on Inform. Theory and its Applications, ISITA 2004, Parma, Italy, October 2004, pp. 1503 - 1506
  D. Korobkov, V. Potapov, V. Sidorenko
  
• "Decomposition of Multidimensional Gray Codes". Ninth Int. Workshop on Algebraic and combinatorial coding theory, Proceedings, June 19-25,2004, Kranevo, Bulgaria, pp. 355 - 361
  V. Sidorenko, M. Starykh
  
• "Multicarrier Spread Spectrum: A Coding Perspective". Eighth IEEE International Symposium on Spread Spectrum Techniques and Applications, 30 August 2 September 2004, Sydney, Australia, pp. 61 - 66
  B. Baumgartner, V. Sidorenko, M. Bossert
  
• "Polyalphabetic codes". Ninth Int. Workshop on Algebraic and combinatorial coding theory, Proceedings, June 19-25,2004, Kranevo, Bulgaria, pp. 171 - 177
  E. Gabidulin, M. Bossert
  
• "Encoding and distance estimation of product convolutional codes". Proceedings of Intern. Symposium on Inform. Theory, Adelaide, Australia, 4-9 September, 2005, pp. 1063- 1067
  M. Bossert, C. Medina, V. Sidorenko
  
• "Error and Erasure Correction of Interleaved Reed-Solomon Codes". Proceedings, Workshop on Coding and Criptography 2005, WCC2005, March 14-18, Bergen, Norway, pp. 20-29
  G. Schmidt, V. R. Sidorenko, and M. Bossert
  
• "Interleaved Reed-Solomon codes in concatenated code designs". In: Proc. of IEEE Inform. Theory Workshop on Coding and Complexity, 28th August - 1st September, 2005, Rotorua, New Zealand, pp. 187-191
  G. Schmidt, V. R. Sidorenko, and M. Bossert
  
• "On polyalphabetic block codes". In: Proc. of IEEE Inform. Theory Workshop on Coding and Complexity, 28th August - 1st September, 2005, Rotorua, New Zealand, pp. 207-210
  V. Sidorenko, G. Schmidt, E. Gabidulin, M. Bossert, V. Afanassiev
  
• "Decoding Reed-Solomon codes beyond half the minimum distance using shift register synthesis". Intern. Symposium on Inform. Theory, Seattle USA, July 9-14, pp. 459-463, 2006
  G. Schmidt, V R. Sidorenko, M. Bossen
  
• "Error Exponents for Product Convolutional Codes". Problems of Information transmission, Vol. 42, No. 3, pp. 3-20, 2006
  C. Medina, V. Sidorenko, V. Zyablov
  
• "Finding Distance Profiles of Unequal Error Protection Codes". Joint 4th International Symposium on Turbo Codes and Related Topics, and 6th International ITG Conference on Source and Channel Coding, Munich, Germany, 3-7 April, 2006
  M. Bossert, V. Sidorenko, V. Zyablov
  
• "Heterogeneous interleaved Reed-Solomon code designs". In: Proc 10th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 230-233, Zvenigorod, Russia, September 2006
  G. Schmidt, V.R. Sidorenko, and M. Bossert
  
• "Multi-Sequence Linear Shift-Register Synthesis: The Varying Length Case". Intern. Symposium on Inform. Theory, Seattle USA, July 9-14, pp. 1738-1742,2006
  G. Schmidt, V. R. Sidorenko
  
• "Polyalphabetic Reed-Solomon Codes in OFDM Transmission". 11th International OFDM-Workshop 2006 (InOWo'06) August 30th - 31st, Hamburg, Germany
  V. Sidorenko, A. Mambetaliev, M. Bessert, S. Edinger, N. Fliege
  
• "Enhancing the correcting radius of interleaved Reed-Solomon decoding using syndrome extension techniques". ISIT2007, Nice, France, June 24-29,2007, pp. 1341-1345
  G. Schmidt, V.R. Sidorenko, and M. Bossert
  
• "Finding Distance Profiles of Unequal Error Protection Block Codes". Eur. Trans. Telecomms, 2007, 18, 541-546
  M. Bessert, V. Sidorenko, V. Zyablov
  
• "From Block to Convolutional Codes using Block Distances". ISIT2007, Nice, France, June 24 June 29, 2007, pp. 2331-2335
  V. Sidorenko, C. Medina, M. Bossen