Improved decoding of Shannon-Kotel´nikov mappings

In some applications the transmission of discrete-time but continuous-amplitude (or multilevel) source symbols is required which might be more bandwidth efficient than conventional digital transmission. An appropriate method is to apply a source channel mapping (SCM) of M source symbols to N channel symbols. A geometrical approach for SCM has been introduced by Shannon and Kotel´nikov (Shannon-Kotel´nikov mappings). These systems are used to map M continuous-amplitude and discrete-time source symbols to N continuous-amplitude and discrete-time channel symbols without the intermediate step of a binary representation. These schemes are usually decoded using a maximum likelihood (ML) decoder which leads to optimum results in the mean square error sense for very good channels, but is suboptimal for noisy channels. In this paper the performance of an improved decoder, the minimum mean square error (MMSE) decoder is assessed. As a special case of a 1:2 expansion case (rate 1/2) Shannon-Kotel´nikov mapping, the Archimedes spiral is considered. The properties of the ML and MMSE decoder are examined and a graphical interpretation of the superior performance of the MMSE decoder is given. Furthermore, the robustness of the MMSE decoder w.r.t. an inaccurate estimation of the channel quality is determined. The concepts of the MMSE decoder which lead to a superior performance to the ML decoder can be generalized and applied to all Shannon-Kotel´nikov mappings.