1726 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 6, NOVEMBER 2004 Transmit Diversity for Arrays in Correlated Rayleigh Fading Cornelius van Rensburg, Member, IEEE, and Benjamin Friedlander, Fellow, IEEE Abstract—Transmit diversity is usually presented for the case of independently faded channels. In this paper the structure of a linear transmitter that can be optimized for a Rayleigh-fading environment in which the fading may be correlated is derived. The transmitter achieves the best mix of array gain—obtained by beamforming, and diversity gain—obtained by using multiple transmit beamformers and space-time coding. The authors use a multiinput single output (MISO) transmitter and receiver struc- ture to present a detailed performance analysis, that shows the array gain versus diversity gain tradeoff as the fading correlation changes. This analysis is validated by simulation results. Index Terms—Array processing, communication, estimation and detection, fading. I. INTRODUCTION I N this paper the authors study the optimal use of an antenna array at a cellular base station (BS), while the mobile station (MS) has a single antenna, also called a multiinput single output (MISO) system. The authors are interested in the use of a com- bination of beamformers and transmit diversity. Transmit diver- sity systems are typically optimal in the rich scattering environ- ments existing in indoor or pico-cellular environments, which means that the fading between the antenna elements are inde- pendent. Thus diversity gain is maximized when the variance of the receiver SNR is minimized. On the other hand, antenna gain is maximized by doing beamforming, typically in line of sight, or low scattering environments where the fading is corre- lated. Maximizing antenna gain means that the authors are max- imizing the average SNR at the receiver. They are interested in the performance of these technologies as the fading correlation changes from perfectly correlated to completely independent. Some of the recent work in using arrays in the downlink ad- dressed optimal solutions for particular environments. Transmit diversity systems which are optimal in independently faded en- vironments have been proposed by [1]–[5] and others. In [1] the authors introduced the concept of space time block codes (STBC), which includes the Alamouti codes as a special case. The same information is transmitted on all the antenna elements, but is multiplexed in a different way on each antenna. In the Manuscript received September 30, 2003; revised April 24, 2004, July 9, 2004, and July 12, 2004. The work of B. Friedlander was supported in part by the Office of Naval Research under Contracts N00014-00-1-0336 and N00014-01-1-0075 and by the National Science Foundation under Grant 0112508. C. van Rensburg is with Samsung Telecommunications America, Dallas, TX 75248 USA (e-mail: cdvanren@ece.ucdavis.edu). B. Friedlander is with the University of California, Santa Cruz, CA 95064 USA (e-mail: friedlan@cse.ucsc.edu). Digital Object Identifier 10.1109/TVT.2004.836959 space time spreading (STS) system proposed by [2] the infor- mation is code multiplexed on different antennas for a CDMA system. Both systems introduce diversity at the cost of losing array gain. In this paper, frequency division duplexing (FDD) systems is considered, where the up and downlink channels are fading in- dependently. Therefore, the BS is unable to estimate the down- link channel, which is needed for designing the downlink beam- former. To overcome this problem, some authors such as [6] proposed a feedback system where this channel information is transmitted back to the BS. This feedback is not desirable in practice because it uses up some of the uplink capacity. More re- cently, [7] proposed a method that combines beamforming and STS for a CDMA2000 system. They used two arrays, each with two elements, and did beamforming with each of the two ele- ment arrays. Others, like [8] also proposed combining space- time codes and beamformers for a MIMO system. They con- sidered a scenario of a number of distinct point sources, and showed through simulations how the frame error rate improves as they use different sized space-time codes. Lately, some per- formance analyzes of MIMO and MISO systems was published by [9]–[12]. The authors will elaborate on this work in the dis- cussion section, since these results agree with the results in this paper. In this paper a linear open-loop transmit diversity system which does not require feedback is described. This system com- bines STBCs with multiple beamformers, and the performance can be optimized for a given fading environment. Since the STBC system has pure diversity gain and a beamformer has pure array gain, the authors strive to find a system with an optimum diversity/antenna gain ratio. Indeed our system changes from a pure STBC transmit diversity system if the channels are inde- pendently faded, to some hybrid system for partially correlated fading channels, to a single beamformer for perfectly correlated fading channels. Based on the available knowledge at the trans- mitter, consider the following scenarios: 1) The channel statistics are known at the transmitter. 2) The channel statistics are unknown at the transmitter. 3) The channel is known at the transmitter. This represents the ideal case where maximum ratio combining (MRC) is possible and will have the best possible performance. Scenarios 1) and 2) are useful in demonstrating the antenna gain versus diversity gain tradeoff which happens here. Scenario 3) is possible either in a time division duplexing (TDD) system, or by using excessive feedback from the MS and is therefore unrealistic from an FDD implementation perspective, but is in- teresting as a reference. In all the scenarios, assume that the channel is known perfectly at the receiver, however, simulation 0018-9545/04$20.00 © 2004 IEEE