Optimum waveforms subject to both energy and peak–value constraints

For any stable linear system with kernel K(t, τ), we derive a formula for the input waveform W(t) which maximizes |R(T)| the absolute value of the output waveform at a prescribed time instant T, subject to the constraints that |W(t)| ≤ A and ∫-∞^∞[W(t)]²dt ≤ E, where A and E are prescribed positive constants. The input and output are related by the formula R(t) = ∫-∞^∞K(t, τ)W(τ)dτ. Roughly, the optimum W(t) can be found by scaling the system kernel function appropriately and then "clipping" it at the value A. The problem and the solution are closely related to the matched filter problem of radar theory and the "bang-bang" problem of optimal control theory and their solutions. The well-known results of these problems appear as special cases of our solution when A or E, respectively, becomes sufficiently large. Theoretically, this problem is of interest because of the incompatible nature of the constraints. That is, the combination of constraints prevents one from using standard optimization tools. Practically, in addition to its direct application in signal design, our solution can also be used indirectly, since it provides a standard of performance for more realistic situations where additional constraints, such as range resolution in radar, are important. The solution is justified in three different ways, providing a comparison of optimization techniques and an example of the Pontryagin Maximum Principle.