Optimal waveform design via control theoretic concepts

This paper presents the formulation of a class of optimal signal design problems; in particular, the design of optimal “modulation≓ waveforms (under amplitude and energy constraints) to be used for the optimal estimation or prediction of the state of a linear dynamical system in the presence of Markov and white noise. The Kalman-Bucy theory is used to formulate an optimization problem in which the “state of the plant≓ is given by the elements of covariance matrices (which satisfy matrix Riccati differential equations), the “control≓ is related to the modulation signal, and the “cost≓ is a linear functional of the covariance matrix. The problem statement is in such a form so that the Maximum Principle of Pontryagin can be applied to obtain the necessary conditions and it is used to derive the so-called “on-off≓ principle. The results prove that under total energy and peak amplitude constraints, the optimal signal, as a function of time, must alternate between its peak-power and its zero-power levels. A set of matrix differential equations—with split boundary conditions—is also derived; the numerical solution of this set of equations can yield the explicit time-dependence of the optimal waveform