IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 6, JUNE 2003 911 On Diagonal Algebraic Space–Time Block Codes Mohamed Oussama Damen, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE Abstract—Theoretical and practical aspects of diagonal alge- braic space–time block codes over transmit and receive antennae are examined. These codes are obtained by sending a rotated version of the information symbols over the principal di- agonal of the space–time matrix over transmit antennae and symbol periods. The output signal-to-noise ratios of two predecoding filters and two decoding algorithms are derived. Analysis of the information loss incurred by using the codes considered is used to clarify their structures, and the expected performances. Different algebraic real and complex rotations presented in the literature are analyzed and compared as re- gards the achieved coding gains, the complexities, performances, and peak-to-mean envelope power ratios. Index Terms—Capacity, diversity methods, maximum-likeli- hood decoding (MLD), multiple-input/multiple-output (MIMO) systems, space–time (ST) codes. I. INTRODUCTION H IGH-RATE data transmission achieved using multiple antennae with diversity techniques has been the subject of many works in the last five years [1]–[6]. Although diversity techniques are a mature topic (see [7] and references therein), fewer results pertaining to diversity techniques applied to sufficiently spaced transmit antennae have been reported. Space–time (ST) codes were proposed by Tarokh et al. in [3] in order to exploit the transmit diversity in a multiantenna system while transmitting at high data rates. Block ST codes based on orthogonal design (OD) were proposed in [8]. They achieve maximum transmit diversity and have the great advantage of optimal linear processing decoding. These codes were proved to maximize the output signal-to-noise ratio (SNR) when the receiver uses linear decorrelators [9]; however, they have small data rates compared to the capacity of the multiantenna system [1]. In [10], DaSilva and Sousa proposed a diagonal scheme over transmit antennae that transmits the components of rotated -dimensional binary phase-shift keying (BPSK) modulations over the different transmit antennae. The rotations used in [10] were optimized either by exhaustive search or by the gradient method over the BPSK constellation in order to obtain constellation diversity. In [6], Damen et al. proposed an approach that generalized [10] to achieve transmit diversity by using diagonal block ST codes based on algebraic rotations from [11]–[13] which transmit at a rate of one symbol Paper approved by I. Lee, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received November 5, 2001; revised August 8, 2002 and October 22, 2002. This paper has been presented in part at the European Wireless Conference, Florence, Italy, February 25–28, 2002. The authors are with the Department of Electrical and Computer Engi- neering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: damen@ee.ualberta.ca; beaulieu@ee.ualberta.ca). Digital Object Identifier 10.1109/TCOMM.2003.813179 per channel use. The so-called diagonal algebraic space–time (DAST) block codes achieve maximum transmit diversity, and their coding gains were optimized over transmit and receive antennae for all pulse-amplitude modulation (PAM) and quadrature amplitude modulation (QAM) constellations on quasi-static and fast-fading channels [6]. They were shown to outperform the OD codes [8] for , with the performance gain increasing as , , and the size of the constellation increase [6]. In this paper, several theoretical and practical aspects of the DAST block codes are examined. The main focus of the paper is to clarify and quantify the structure and the performance of the DAST block codes [6] with respect to different criteria. We derive the output SNR of two predecoding filters of the DAST block codes, the maximum ratio combiner (MRC), and the equal gain combiner (EGC). The outputs of these filters are fed to a DAST decoder, where we consider two classes of al- gorithms, maximum-likelihood (ML) decoding implemented by the sphere decoder [16], [17], and successive interference can- cellation (SIC), a suboptimal decoding represented here by a two-step QR detector [18]. We compute the capacities of the newly precoded channels when using the DAST block codes in closed form, which elucidates the structures and performances of these codes. Finally, we discuss the advantages and disadvan- tages of different real and complex rotations in the literature. II. SYSTEM MODEL The following notations are used in this paper. Matrices and vectors are denoted in boldface. We denote by , , , and , the integer ring, the rational numbers field, the real numbers field, and the complex numbers field, respectively. We denote by , and the Gaussian integer ring and the complex rational field, respectively, where . A linear ST block code is defined as a linear one-to-one cor- respondence that maps the information symbol vector to an matrix , such that one transmits at time over transmit antenna , after normalizing the total transmitted power by . In quasi-static fading where the channel coefficients (gains) are fixed over symbol periods, the received signal over receive antennae and symbol periods is (1) where is the channel matrix with representing the fading between the th transmit and th receive antenna. The fading coefficients are modeled by independent, identi- cally distributed (i.i.d.) zero-mean complex Gaussian random variables with a common variance of 0.5 per real dimension. The additive noise has entries which are modeled 0090-6778/03$17.00 © 2003 IEEE