Inverses of finite group systems

Inverse systems are considered for a class of discrete time-invariant systems that include the finite linear sequential circuits (LSC's). Invertibility results for finite group homomorphic sequential systems (FGHSS's), given by Willsky [8], are extended to include systems with throughput A construction is developed for an
L
-delay inverse of any FGHSS that is invertible with
L\geq0
delays. This inverse is always a discrete time-invariant system. Necessary and sufficient conditions are given for the inverse to consist of only homomorphic maps when the FGHSS state group is abelian, and only homomorphic and antihomomorphic maps when nonabelian. In the abelian case these conditions are necessary and sufficient for the existence of an inverse system of the specified delay that is itself an FGHSS. Invertible FGHSS's can be regarded as a generalization of convolutional encoders, since they include the class of invertible finite LSC's as a proper subclass.