Overlapped Arithmetic Codes

TL;DR

This paper introduces overlapped arithmetic codes, a generalization of traditional arithmetic coding, enabling distributed source coding, joint source-channel coding, and hybrid coding by redefining interval mappings.

Contribution

It proposes a unified framework for generalized arithmetic codes, including overlapped, forbidden, and hybrid codes, expanding their application beyond classical lossless source coding.

Findings

Overlapped arithmetic codes enable distributed source coding.

Forbidden arithmetic codes facilitate joint source-channel coding.

Hybrid codes support combined distributed and joint coding scenarios.

Abstract

Arithmetic codes are usually deemed as the most important means to implement lossless source coding, whose principle is mapping every source symbol to a sub-interval in [0, 1). For every source symbol, the length of its mapping sub-interval is exactly equal to its probability. With this symbol-interval mapping rule, the interval [0,1) will be fully covered and there is neither overlapped sub-interval (corresponds to more than one source symbol) nor forbidden sub-interval (does not correspond to any source symbol). It is well-known that there is a duality between source coding and channel coding, so every good source code may also be a good channel code meanwhile, and vice versa. Inspired by this duality, arithmetic codes can be easily generalized to address many coding problems beyond source coding by redefining the source-interval mapping rule. If every source symbol is mapped to an enlarged sub-interval, the mapping sub-intervals of different source symbols will be partially overlapped and we obtain overlapped arithmetic codes, which can realize distributed source coding. On the contrary, if every source symbol is mapped to a narrowed sub-interval, there will be one or more forbidden sub-intervals in [0, 1) that do not correspond to any source symbol and we obtain forbidden arithmetic codes, which can implement joint source-channel coding. Furthermore, by allowing the coexistence of overlapped sub-intervals and forbidden sub-intervals, we will obtain hybrid arithmetic codes, which can cope with distributed joint source-channel coding.