Impact of the sampling theorem interpretations on digitization and reconstruction in SDRs and CRs

Sampling and reconstruction (S&R) are used in virtually all areas of science and technology. The classical sampling theorem is a theoretical foundation of S&R. However, for a long time, only sampling rates and ways of the sampled signals representation were derived from it. The fact that the design of S&R circuits (SCs and RCs) is based on a certain interpretation of the sampling theorem was mostly forgotten. The traditional interpretation of this theorem was selected at the time of the theorem introduction because it offered the only feasible way of S&R realization then. At that time, its drawbacks did not manifest themselves. By now, this interpretation has largely exhausted its potential and inhibits future progress in the field. This tutorial expands the theoretical foundation of S&R. It shows that the traditional interpretation, which is indirect, can be replaced by the direct one or by various combinations of the direct and indirect interpretations that enable development of novel SCs and RCs (NSCs and NRCs) with advanced properties. The tutorial explains the basic principles of the NSCs and NRCs design, their advantages, as well as theoretical problems and practical challenges of their realization. The influence of the NSCs and NRCs on the architectures of SDRs and CRs is also discussed.