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

The problem of maximizing the output energy of a linear time-invariant channel, given that the input signal is time and amplitude limited, is considered. It is shown that a necessary condition for an input μ to be optimal, assuming a unity amplitude constraint is that it satisfy the fixed-point equation=sgn [F(μ)], where the functional F is the convolution of μ with the autocorrelation function of the channel impulse response. It is also shown that all solutions to this equation for which |μ|=1 almost everywhere correspond to local maxima of the output energy. Iteratively recomputing μ 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 low-pass 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. 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