A model representative of any time-sampled, real-valued input and output digital system without feedback from system input to output (there may be internal feedback loops) is presented, and properties that are necessary for an optimum (nonlinear) digital system are developed. The model itself consists of a cascade of three mapping operations that successively transform the real-valued input sequence into an M-ary valued digital sequence, thence into anN-ary valued digital sequence, and finally into a real-valued output sequence. The operating routine and optimality criterion function chosen permit us to consider the problems of prediction, interpolation or synchronous operation as well as a wide variety of loss functions. The derived properties are discussed with particular regard to the quadratic loss function and synchronous or interpolatory operation and then applied to the determination of several optimum systems. Optimum nonlinear amplitude quantizers, pulse code and delta modulation systems, and stepped controllers are described. In particular, conditions are given under which delta modulation is superior to pulse code modulation and some remarks are made concerning suboptimal delta modulation systems and their relation to previous work.