Waveform Design with Time and Frequency Constraints for Optimal Detection of Elastic Objects

We extend a recent method by Kay that maximizes the probability of detecting an elastic object in the presence of Gaussian reverberation and additive Gaussian interference. Kay's solution specifies the spectral magnitude for the optimal transmit waveform, and hence there is an unlimited number of "optimal" waveforms that can be transmitted, all with the same spectral magnitude but differing in terms of time domain characteristics such as duration and peak power. We extend Kay's approach in order to obtain a unique optimal waveform by incorporating time-domain constraints, via two optimization problem formulations. One approach yields a waveform that preserves the optimal spectral magnitude while achieving the minimum temporal duration. The second complementary approach considers temporal concentration rather than duration, and yields a waveform that, depending on the degree of concentration imposed, achieves the optimal the spectral magnitude to varying degrees.