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

Energy-Efficient Full Diversity Collaborative Unitary Space-Time Block Code Design via Unique Factorization of Signals