Minimal Memory Inverses of Linear Sequential Circuits

A necessary and sufficient condition for a linear sequential circuit to possess a feedforward inverse is given. This condition for existence of a feedforward inverse is given in terms of the Markov parameters rather than the minors of the transfer function matrix of the linear sequential circuit as has been the case in previous studies on this problem. An upper bound on the required memory of a feedforward inverse for a linear sequential circuit is established in terms of n, the dimension of the state transition matrix and Lo, the inherent delay of the linear sequential circuit. When the original linear sequential circuit is itself feedforward, a stronger upper bound on the memory of a feedforward inverse is established in terms of u, the input memory, k, the number of inputs, and Lo. Both upper bounds on the required inverse memory are shown to be tight in the sense that for each set of possible values of the parameters entering the bound there exists at least one linear sequential circuit whose inverse requires input memory equal to the bound. Finally, a practical and simple algorithm is given to construct a minimal memory inverse.