  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.
