Constructive algorithms for determining inverses and syndrome matrices of multidimensional convolutional encoders using the Gröbner basis approach

A representation of an m-D convolutional encoder is analogous to the representation of a transfer function for a MIMO m-D FIR system. The encoder matrix is usually not square and thus finding its inverse typically employs the Moore-Penrose generalized inverse. However, the result may not be FIR (polynomial matrix) even if the generator matrix is a polynomial. In this paper a constructive algorithm based on the usage of Grobner basis is presented along with detailed examples. The result obtained can be parameterized to cover the class of all possible FIR inverses. In addition, by using the computation method of syzygy with the Grobner basis module, the syndrome matrix for a given m-D convolutional encoder is shown.