Equivalence of nonlinear shift-registers

Two forms of nonlinear-feedback shift-registers are considered. In the Type-I register, the feedback output is added to the shift-register contents at an arbitrary number of stages. In the type-II register, the feedback is input to the first stage only. It is shown that for every Type-I register there is an equivalent Type-II register in the sense that the autonomous state diagrams differ only by a labelling of the states. Moreover, the mapping between equivalent states can always be chosen to be a linear transformation. This theorem is a well-known result in the theory of linear-feedback shift-registers and is thus seen to apply unchanged to the nonlinear case.