 
Summary: Pairwise Optimization of Modulation
Constellations for NonUniform Sources
Brendan Moore, Glen Takahara and Fady Alajaji
The design of twodimensional signal constellations for the transmission of binary nonuniform memoryless sources over additive white Gaussian
noise channels is investigated. The main application of this problem is the implementation of improved constellations where transmitted data is highly
nonuniform. A simple algorithm, which optimizes a constellation by rearranging its points in a pairwise fashion (i.e., two points are modified at a time,
with all other points remaining fixed), is presented. In general, the optimized constellations depend on both the source statistics and the signaltonoise
ratio (SNR) in the channel. We show that constellations designed with source statistics considered can yield symbol error rate (SER) performance that is
substantially better than rectangular quadrature amplitude modulation signal sets used with either Gray mapping or more recently developed maps. SER
gains as high as 5 dB in Eb/N0 SNR are obtained for highly nonuniform sources. Symbol mappings are also developed for the new constellations using
a similar pairwise optimization method whereby we assign and compare a weighted score for each pair. These maps, when compared to the mappings used
in conjunction with the standard rectangular QAM constellation, again achieve considerable performance gains in terms of bit error rate (BER). Gains
as high as 4 dB were achieved over rectangular QAM with Gray mapping, or more than 1 dB better than previously improved mappings. Finally, the
uncoded pairwise optimized system is compared to a standard tandem (separate) source and channel coding system. Although neither system is universally
better, the uncoded system with optimized constellations outperforms the tandem coding system for lowtomid SNRs. Performance/complexity tradeoffs
between the two systems are also discussed.
I Introduction
For uniformly distributed sources, rectangular quadrature amplitude
modulation (QAM) using Gray mapping is known to perform well, and
is shown as optimal in terms of bit error rate (BER) for high enough
