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 (S-K) mappings is discussed. S-K mappings are continuous, or piecewise smooth direct source-to-channel mappings operating on amplitude-continuous and discrete-time signals, and they encompass 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 variable noise level at both low and arbitrary complexity and delay. However, a general theory explaining their performance and behaviour, as well as guidelines on how to construct well-performing mappings, do not exist. Therefore, such mappings are often based on educated guesses inspired by configurations that are known in advance to produce good solutions through numerical optimization methods. The objective of this paper is to develop a theoretical framework for analysis of analog- or hybrid discrete-analog S-K mappings which enables calculation of distortion when applying them on point-to-point links, reveal more about their fundamental nature, and provide guidelines for their construction at low as well as arbitrary complexity and delay. Such guidelines will likely help constrain solutions to numerical approaches and help explain why deep learning approaches obtain the solutions they do. The overall 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 mappings based on surfaces which are chosen, or constructed, based on the provided theory.