Signal-Noise Ratio Maximization Using the Pontryagin Maximum Principle

The Pontryagin maximum principle may be considered to be a generalization of methods of calculus of variations that permits solution of optimization problems with inequality contraints. During the last few years, it has been extensively used to attack control theory problems. The use of the principle to solve signal optimization problems is introduced in Ref. 1. The maximum principle is briefly discussed in connection with wave-form optimization in Ref. 2. The purpose of this present paper is to develop techniques for the application of the maximum principle, in particular to problems of signalnoise ratio maximization. We shall show how the maximum principle may be used to solve some problems with inequality constraints (e.g., the amplitude of a signal may be constrained to be less than or equal to some maximum value) which were heretofore considered intractable. We shall also show how a problem, solvable by other methods, may be 473 474 T H E BELL SYSTEM TECHNICAL JOURNAL, MARCH 1966 very conveniently attacked with the formalism of the maximum principle. It is interesting to note that the maximum principle is, in a sense, more applicable to communication theory than to control theory for which it was originally developed (this is also pointed out in Ref. 1). The maximum principle yields a function of time to maximize a functional subject to constraints and for prescribed initial conditions. The answer to most communication theory problems is a function of time. On the other hand, in control problems, the function of time for specific initial conditions is called an "open loop" solution.