Maximizing the output energy of a linear channel with a time- and amplitude-limited input

The following problem is considered: Maximize the output energy of a linear time-invariant channel, given that the input signal is time and amplitude limited. It is shown that a necessary condition for an input U to be optimal, assuming a unity amplitude constraint, is that it satisfy the fixed-point equation U = sgn [F(u)], where the functional F is the convolution of U with the autocorrelation function of the channel impulse response. It is also shown that all solutions to this equation for which I U 1 = 1 almost everywhere correspond to local maxima of the output energy. Iteratively recomputing U from the fixed-point equation leads to an algorithm for finding local optima. Numerical results are given for the cases where the transfer function is ideal lowpass, and has two poles. These results support the conjecture that in the ideal low-pass case the optimal input signal is a single square pulse. Often, several local optima are found by the iterative algorithm, and the global optimization problem appears to be computationally intractable. A generalization of the preceding fixed-point condition is also derived for the problem of maximally separating N outputs of a discrete-time, linear, time-invariant channel, assuming the inputs are constrained in time and amplitude.