Classification of Unique Mappings for 8PSK Based on Bit-Wise Distance Spectra
Artikel i vetenskaplig tidskrift, 2009

The performance of bit-interleaved coded modulation (BICM) with (or without) iterative decoding (ID) is significantly influenced by the mapping of bits to the symbol constellation. Our main objective in this paper is to develop a systematic design approach for BICM-ID schemes, ensuring the best possible performance with iterative decoding. Although useful mappings for BICM-ID have been found based on various search strategies, no attempt has been made to systematically enumerate and classify all unique mappers for a given constellation. As the basis for a systematic enumeration and classification, we define the average bit-wise distance spectrum for a mapping from bits to symbols. Different bit-wise distance spectra are derived assuming no prior information or full prior information, respectively. The bit-wise distance spectra determine corresponding bit-wise error probability and bit-wise mutual information. The latter allows us to use the classification of mappings with unique bit-wise distance spectra to also classify mappings with unique extremal points in the corresponding extrinsic information transfer (EXIT) curves. As an example of our approach, we classify 8PSK mappings into 86 classes of unique mappings according to bit-wise distance spectra. The classification can be used to significantly reduce the complexity of the search for suitable mappers for BICM-ID. For 8PSK and a given encoder, only 86 different mappings need to be, investigated. As examples of the systematic design approach, the best 8PSK mappings for minimizing the convergence threshold are found for concatenation with the rate 1/2 (5, 7)(8) and (133, 171)(8) convolutional codes, and the rate 1/2 UMTS turbo code with identical constituent convolutional codes (15/13)(8).

gray code


error rate


interleaved coded modulation

bit-interleaved coded modulation


concatenated codes



8PSK mappings

EXIT chart

error probability

mutual information

m-ary psk

bit-wise distance



Fredrik Brännström

Chalmers, Signaler och system, Kommunikation, Antenner och Optiska Nätverk

Lars K. Rasmussen

University of South Australia

Kungliga Tekniska Högskolan (KTH)

IEEE Transactions on Information Theory

0018-9448 (ISSN)

Vol. 55 3 1131-1145


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