Unitary signal constellations for differential space--Time modulation with two transmit antennas: Parametric codes, optimal designs, and bounds

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 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. In this paper, 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 leads to a five-signal constellation with the largest possible diversity product and a 16-signal constellation with the largest known diversity product. Although the parametric code of size 16 is not a group by itself, we show that it is a subset of a group of order 32. 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 5 and 9 through 16 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 6, 7, and 8 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 Manuscript received December 22, 2000; revised August 23, 2001. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grants F49620-00-1-0086 and F49620-02-1-0157, and by the National Science Foundation under Grants MIP-9703377 and CCR-0097240. The material in this paper was presented in part at the 34th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, October 29–November 1, 2000 and the IEEE International Symposium on Information Theory, Washington, DC, June 24–29, 2001. X.-B. Liang was with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA. He is now with the Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803 USA. X.-G. Xia is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: [email protected]). Communicated by G. Caire, Associate Editor for Communications. Publisher Item Identifier 10.1109/TIT.2002.800498. product and the largest possible diversity sum that unitary signal constellations of any size can achieve.