In the case of band-limited signals, the sampling theorem permits us to replace analytic operations with algebraic operations. We are then able to discuss problems of measurement of information, coding, and transmission over noisy channels in terms of discrete samples, rather than continuous time functions. The design of the optimum linear filter reduces from a very difficult analysis problem involving spec- trum factorizatlon to a straightforward problem of solving a set of simultaneous linear equations. Unless we are interested in the most economical implementation, it is not even necessary to solve the equations. since a synthesis procedure involving only simple functions of the correlation functions is available. When extended to the general nonlinear case, the design is still specified by a set of simultaneous algebraic equations, but the labor of solution grows very rapidly. It is proposed to short circuit this labor by building a learning filter which in effect designs itself. A training period in which the adjust- ments are automatically optimized precedes the use period. By modifying the training program, it is possible that the filter could be taught to recognize specific signals, including, perhaps, certain speech sounds.