Optimized shaping and trade-off of superoscillating pulses

In this article, we consider a special family of pulses, which are a part of a band limited signal and are ‘too narrow’ considering the band limit of the signal. We dub such pulses ‘superoscillating pulses’ although they can be seen at best as half an oscillation. While some of our results are of a more generic nature, the article is devoted to the optimization of superoscillating pulses, The first step consists of approximating a given target signal by a band limited signal in a fixed time interval. The signals to be approximated, exhibit in that interval features that seem to involve frequencies higher than the band limit of the approximant. We define the mean square relative error (MSRE) as a measure of the adherence of the approximant to the original signal in the chosen interval. We find that the minimization of the MSRE conflicts with the necessity to minimize the energy (or power) expense of the superoscillating signal. We obtain the trade-off relation between optimal energy expense and the MSRE for a family of pulses. This makes it clear that within that family, there exists a specific pulse shape that is better approximated by a superoscillating pulse. Finally, we show how to construct a yield optimized superoscillating pulse that, within a given time interval, has a prescribed narrow width without resorting to a target signal at all.