  A general method of coding over expansion is proposed,which allows one to
reduce the highly non-trivial problems of coding over analog channels and
compressing analog sources to a set of much simpler subproblems, coding over
discrete channels and compressing discrete sources. More specifically, the
focus of this paper is on the additive exponential noise (AEN) channel, and
lossy compression of exponential sources. Taking advantage of the essential
decomposable property of these channels (sources), the proposed expansion
method allows for mapping of these problems to coding over parallel channels
(respectively, sources), where each level is modeled as an independent coding
problem over discrete alphabets. Any feasible solution to the resulting
optimization problem after expansion corresponds to an achievable scheme of the
original problem. Utilizing this mapping, even for the cases where the optimal
solutions are difficult to characterize, it is shown that the expansion coding
scheme still performs well with appropriate choices of parameters. More
specifically, theoretical analysis and numerical results reveal that expansion
coding achieves the capacity of AEN channel in the high SNR regime. It is also
shown that for lossy compression, the achievable rate distortion pair by
expansion coding approaches to the Shannon limit in the low distortion region.
Remarkably, by using capacity-achieving codes with low encoding and decoding
complexity that are originally designed for discrete alphabets, for instance
polar codes, the proposed expansion coding scheme allows for designing
low-complexity analog channel and source codes.
