Shannon-Kotel'nikov Mappings for Analog Point-to-Point Communications

In this paper an approach to joint source-channel coding (JSCC) named Shannon-Kotel'nikov mappings (S-K mappings) is presented. S-K mappings are continuous, or piecewise continuous direct source-to-channel mappings operating directly on amplitude continuous and discrete time signals. Such mappings include several existing JSCC schemes as special cases. Many existing approaches to analog- or hybrid discrete analog JSCC provide both excellent performance as well as robustness to variations in noise level. This at low delay and relatively low complexity. However, a theory explaining their performance and behaviour on a general basis, as well as guidelines on how to construct close to optimal mappings in general, does not currently exist. Therefore, such mappings are often found based on educated guesses inspired of configurations that are known in advance to produce good solutions, combination of already existing mappings, numerical optimization or machine learning methods. The objective of this paper is to introduce a theoretical framework for analysis of analog- or hybrid discrete analog S-K mappings. This framework will enable calculation of distortion when applying such schemes on point-to-point links, reveal more about their fundamental nature, and provide guidelines on how they should be constructed in order to perform well at both low and arbitrary complexity and delay. Such guidelines will likely help constrain solutions to numerical approaches and help explain why machine learning approaches finds the solutions they do. This task is difficult and we do not provide a complete framework at this stage: We focus on high SNR and memoryless sources with an arbitrary continuous unimodal density function and memoryless Gaussian channels. We also provide example of mappings based on surfaces which are chosen based on the provided theory.