Convolutional Transformation and Recovery of Binary Sequences

Abstract—This paper analyzes invertible length-preserving convolutional transformations of binary sequences, when the perfect inverse feedback transducer is replaced by a finite feedforward (i. e., convolutional) transducer which represents an approximation of the former. This replacement eliminates the error propagation effect, but the finiteness of the "inverse" transducer (decoder) results in a restriction on the input sequences, for which exact replication is achieved in the two-way transduction (transformation and recovery). The entity of the input restriction can be taken as a measure of performance for sets of transformations, and can be described in terms of the upper bound of the entropies of the binary sources which are "matched" to the system (direct and inverse transducers). This bound is clearly the capacity of the system viewed as a noiseless channel. It is shown that, if r is the number of decoder stages, the channel capacity has an asymptotic expression C≃1-Ab
^r
, where the parameters b < 1 and A depend solely upon the structure of the set of resynchronizing states (RS cluster) possessed by the given set of transformations.