  In this paper, a novel concept called a \textit{uniquely factorable
constellation pair} (UFCP) is proposed for the systematic design of a
noncoherent full diversity collaborative unitary space-time block code by
normalizing two Alamouti codes for a wireless communication system having two
transmitter antennas and a single receiver antenna. It is proved that such a
unitary UFCP code assures the unique identification of both channel
coefficients and transmitted signals in a noise-free case as well as full
diversity for the noncoherent maximum likelihood (ML) receiver in a noise case.
To further improve error performance, an optimal unitary UFCP code is designed
by appropriately and uniquely factorizing a pair of energy-efficient cross
quadrature amplitude modulation (QAM) constellations to maximize the coding
gain subject to a transmission bit rate constraint. After a deep investigation
of the fractional coding gain function, a technical approach developed in this
paper to maximizing the coding gain is to carefully design an energy scale to
compress the first three largest energy points in the corner of the QAM
constellations in the denominator of the objective as well as carefully design
a constellation triple forming two UFCPs, with one collaborating with the other
two so as to make the accumulated minimum Euclidean distance along the two
transmitter antennas in the numerator of the objective as large as possible and
at the same time, to avoid as many corner points of the QAM constellations with
the largest energy as possible to achieve the minimum of the numerator. In
other words, the optimal coding gain is attained by intelligent constellations
collaboration and efficient energy compression.
