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

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.