On the labeling of Signal Constellations
Doctoral thesis, 2005
The work in this thesis studies a simple model of a digital communication system in which the transmitter operates on a sequence of information bits by grouping them into blocks of m bits and mapping such blocks onto one out of M = 2m waveforms, which is transmitted over the channel. The receiver recovers the original message by demapping the transmitted signals disturbed by noise back to m-bit blocks using maximum-likelihood symbol detection. Whenever a bit produced by the source is not regenerated correctly at the destination, a bit error is said to have occurred. The presented work addresses certain methods to limit the occurrence of such events. A communication system operating as described above is commonly studied within a geometrical framework in which the transmitted waveforms are represented as vectors in a linear vector space and this set of vectors is referred to as the signal constellation. The labelling of the signal constellation, i.e., the mapping of m-bit blocks onto the signal constellation is of big importance for the resulting bit error probability (BEP). We study the constellation labelling problem for two-dimensional constellations and for systems using PSK, PAM, or QAM; three types of modulation methods that are commonly encountered in the existing literature and in practice. We present a new expression for the exact BEP of systems using coherent symbol detection of M-PSK, which rectifies a common misconception in previously published results found in the literature. We introduce the average distance spectrum (ADS) which allows for a convenient method to analyze the influence of the constellation labelling on the BEP, which leads to new results relating to the commonly encountered binary reflected Gray code (BRGC). In particular, we show that the BRGC is the optimal labelling with respect to minimizing the BEP over a Gaussian channel as long as the BEP is less than a few percent. This result, which has not been proved before, motivates and validates the extensive usage of the BRGC both in theoretical studies and in practice.
Keywords: constellation labelling, bit-to-symbol mapping, optimal labelling, bit error probability (BEP), Gray code, binary reflected Gray code (BRGC), phase-shift keying (PSK), pulse amplitude modulation (PAM), quadrature amplitude modulation (QAM).