On the synthesis of periodic signals by discrete pulse-trains and optimisation techniques

In this paper, the synthesis of discrete sequences capable of representing analog periodic waveforms and of being transformed into them by mere low-pass filtering is considered. The problem is known to be solvable by modulation techniques such as pulse width modulation or Sigma-Delta. Nonetheless, how close to an optimum such techniques can go is an interesting issue to explore, since the design of such modulators relies on assumptions that can only be approximately respected. To this aim, we develop an exact optimisation framework capable of providing the best and most parsimonious coding possible, which is then used as a benchmark. It is shown that the optimisation problem has very strong features, which could potentially be exploited to speed up convergence or to develop efficient heuristics. At the same time it is verified that the Sigma-Delta performance lag with regards to the optimum is generally very little, so that almost no margin exists for alternative heuristics unless the latter are exploited to optimise on merit factors beyond mere SNR.