Advertisement

Wide-Sense Time-Invariant Convolutional Codes in Their Generator Matrix

  • Giovanni CancellieriEmail author
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

A wide-sense time-invariant convolutional code is characterized by more than one interleaved generator polynomial periodically repeating in its generator matrix. The existence of a right-inverse matrix is to be tested here. The encoder circuit is based on more than one shift register in controller arrangement, or in observer arrangement. A recursive systematic solution is possible, and can be calculated by means of the Smith form of the generator matrix. An equivalence between modified lengthened quasi-cyclic codes and a certain class of w.s. time-invariant convolutional codes is demonstrated. The concept of not well-designed convolutional codes is extended to such codes. Their tail-biting version has analogous characteristics as that introduced for s.s. time-invariant convolutional codes. A first conceptual bridge between quasi-cyclic codes and this type of convolutional codes is outlined. It is based on unwrapping the reordered version of the quasi-cyclic code. Finally, state diagrams and trellises are described. Here the computational complexity has to be calculated also taking into account the number of branches entering the same node. Truncated circulants are introduced for describing reordered versions of convolutional codes not in tail-biting form. A scheme of doubly convolutional code is described for the case in which two series of distributed control symbols are adopted.

Keywords

Generator Matrix Code Rate Block Code Convolutional Code Frame Length 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Bossert M (1999) Channel coding for telecommunications. Wiley, WeinheimGoogle Scholar
  2. Cain JB, Clark GC, Geist JM (1979) Punctured convolutional codes at rate (n − 1)/n and simplified maximum likelihood decoding. IEEE Trans Inf Th 25:97–100CrossRefMathSciNetGoogle Scholar
  3. Costello DJ, Pusane AE, Jones CR et al (2007) A comparison of ARA-and protograph-based LDPC block and convolutional codes. Proc Inf Th Appl Workshop, La Jolla (CA), 111–119Google Scholar
  4. Gallager RG (1962) Low-density parity-check codes. IRE Trans Inf Th 8:21–28CrossRefzbMATHMathSciNetGoogle Scholar
  5. Jimenez-Felstrom AJ, Zigangirov KS (1999) Time-varying periodic convolutional codes with low density parity-check matrix. IEEE Trans Inf Th 45:2181–2191CrossRefMathSciNetGoogle Scholar
  6. Johannesson R, Zigangirov KS (1999) Fundamentals of convolutional coding. IEEE Press, New YorkCrossRefGoogle Scholar
  7. Ryan WE, Lin S (2009) Channel codes: classical and modern. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  8. Solomon G, Van Tilborg HCA (1979) A connection between block and convolutional codes. SIAM J Appl Math 37:358–369CrossRefzbMATHMathSciNetGoogle Scholar
  9. Tanner RM (1981) A recursive approach to low complexity codes. IEEE Trans Inf Th 27:2966–2984CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Information EngineeringPolytechnic University of MarcheAnconaItaly

Personalised recommendations