| Differential space-time modulation has been recently proposed in the literature for multiple antenna systems over Rayleigh fading channels, where neither the transmitter nor the receiver knows the fading coefficients. For the practical success of the differential space-time modulation, it has been shown critical to design unitary space-time signal constellations with large diversity product which is a primary property for the signal constellations to have good performance in high signal-to-noise ratio (SNR) scenarios.; We focus on the design of unitary signal constellations for differential space-time modulation with double transmit antennas. By using the parametric form of a two by two unitary matrix, we present a class of unitary space-time codes called parametric codes and show that this class of unitary space-time codes lead to a five-signal constellation with the largest possible diversity product and a sixteen-signal constellation with the largest known diversity product. Although the parametric code of size sixteen is not a group by itself, we show that it is a subset of a group of order thirty-two. Furthermore, the unitary signal constellations of sizes 32, 64, 128, and 256 obtained by taking the subsets of the parametric codes of sizes 37, 75, 135, and 273, respectively, have the largest known diversity products.; We also use large diversity sum of unitary space-time signal constellations as another significant property for the signal constellations to have good performance in low SNR scenarios. The newly introduced unitary space-time codes can lead to signal constellations with sizes of five and nine through sixteen that have the largest possible diversity sums. Subsequently, we construct a few sporadic unitary signal constellations with the largest possible diversity product or diversity sum. A four-signal constellation which has both the largest possible diversity product and the largest possible diversity sum and three unitary signal constellations with the largest possible diversity sums for sizes of six, seven and eight are constructed, respectively. Furthermore, by making use of the existing results in sphere packing and spherical codes, we provide several upper and lower bounds on the largest possible diversity product and the largest possible diversity sum that unitary signal constellations of any size can achieve.; We make use of the theory of real and complex orthogonal designs for developing a fast decoding algorithm for differential space-time detection with two or more than two transmitter antennas.; Finally, we present a theoretical result that there does not exist rate one space-time block code from generalized complex linear processing orthogonal designs for transmission with more than two antennas using any arbitrary complex constellation. |
