On Nonlinear Binary Sequential Circuits and Their Inverses

This paper discusses single-input, single-output binary sequential circuits composed of adders, multipliers and delay units. The principal results are: 1) Finite feedforward circuits are characterized by a response function (rf) which permits the determination of the output for an arbitrary input. 2) Corresponding to every rf is a transfer function (tf) which is obtained by a multidimensional Z-transform of the rf. This tf permits the determination of the response of a feedforward circuit by transform domain techniques. 3) Every tf may be synthesized as a physical circuit. 4) A canonic form is defined for every tf. With this canonic form is identified an implementation which contains a minimum number of delay units and adders. 5) Nonfeedforward circuits are called well defined if they meet certain physically meaningful restrictions. 6) Well-defined circuits may be classified as those which have inverses (class I) and those which do not. Membership in class I is shown to be a property of circuits which established one-one maps of inputs onto outputs. Circuits which do not belong to class I have neither left nor right inverses. 7) An intimate connection between inverses and feedback circuits is established. It is shown that any well-defined feedback circuit and a related circuit in I are mutually inverse. 8) It is shown that the necessary and sufficient condition for a feedback circuit to be well defined is that the circuit in the feedback loop contains delay.