Generator Matrix Approach to s.s. Time-Invariant Convolutional Codes

Abstract

The traditional description of low-rate convolutional codes is presented. They are interpreted as strict-sense time-invariant in their generator matrix, where an interleaved generator polynomial shifts by a period any time. State diagrams are depicted with reference to the encoder circuit constructed by means of a shift register in controller arrangement. The minimum distance is calculated in this state diagram. Code puncturation is adopted in order to obtain higher code rates. A recursive systematic encoder circuit is described. An important equivalence between modified lengthened cyclic codes and s.s. time-invariant convolutional codes is demonstrated. A first conceptual bridge between cyclic block codes and convolutional codes is presented, showing that the interleaved generator polynomials of many interesting convolutional codes coincide with the generator polynomials of well-known cyclic block codes, mainly Hamming codes. A catastrophic convolutional code is described. In its systematic version it is no longer catastrophic but remains a not well-designed convolutional code, characterized by a more than linear growth in the number of low-weight code frames with the number of periods in the frame. A tail-biting arrangement is introduced, in order to pay a null cost in correct frame termination, so the asymptotic code rate can become the true code rate on whichever frame length. Nevertheless, the frame weight distribution may be penalized when the frame length is too short, especially if the convolutional code is not well designed. Finally, the trellis of a convolutional code is constructed, where the computational complexity of a soft-decision decoding, performed according to Viterbi algorithm, can be easily evaluated. In this sense, the generator constraint length expresses a very important parameter for the design of such type of codes.