618 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 2, FEBRUARY 2008 Optimal Transmission Strategies for Rayleigh Fading Relay Channels Yonglan Zhu, Student Member, IEEE, Yan Xin, Member, IEEE, and Pooi-Yuen Kam, Senior Member, IEEE Abstract—In this paper, a decode-and-forward (DF) single relay model in a Rayleigh fading environment is considered. The outage probability and ergodic rate for this model are derived. With the objective of either minimizing the outage probability or maximizing the ergodic rate, optimal and approximately optimal transmission strategies are developed through selecting appropriate transmit signaling and/or spatial power allocation. The derived transmission strategies only require the knowledge of the second-order statistics of the channels at the transmitters, which can be readily acquired in practice. Simulation results demonstrate that the optimal transmit signaling and/or spatial power allocation can offer considerable performance improve- ments over the equal power allocation strategy. Index Terms— Decode-and-forward, ergodic rate, outage prob- ability, power allocation, relay channels. I. I NTRODUCTION I N wireless networks, the cooperative relaying transmission is an effective technique to improve power efficiency, enhance network coverage, and mitigate detrimental fading effects of wireless channels. The study of such a transmission technique under various settings, especially in relay channels, has received a considerable amount of research attention [1]– [6]. Until now, the capacity of the general relay channels remains unknown except for some special cases [1]. Alter- natively, the highest information rates achieved by various relaying protocols such as decode-and-forward (DF), amplify- and-forward (AF), and compress-and-forward (CF) have been extensively investigated [1], [4], [6]. In fading relay channels, the typical information-theoretic performance measures are the outage probability and the ergodic rate, respectively, for nonergodic and ergodic fading channels. These performance measures generally depend on the statistical correlation of the signals transmitted from source and relay, as well as the spatial power allocation between the two nodes. It was shown in [6] that for Rayleigh fading DF relay channels, the transmit signals from source and relay, which maximize the ergodic rate, are statistically independent. In a low signal-to- noise ratio (SNR) regime, a spatial power allocation strategy that maximizes the upper bound of the ergodic capacity was derived in [7], [8]. In [9], several power allocation strategies Manuscript received August 17, 2006; revised September 9, 2007; accepted October 22, 2007. The associate editor coordinating the review of this paper and approving it for publication is A. Stefanov. This paper was presented in part at the 40th Annual Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, October 2006, and at the International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hon- olulu, HI, USA, April 2007. The authors are with the Department of Electrical and Computer En- gineering, National University of Singapore, Singapore 117576 (e-mail: {zhuyonglan, elexy, elekampy}@nus.edu.sg). Digital Object Identifier 10.1109/TWC.2008.060595. that maximize lower or upper bounds of ergodic capacity were developed for various settings. Different from the previous work [7]–[9] which emphasizes primarily on optimizing er- godic performance measures, a recent paper [10] proposed a spatial power allocation strategy that optimizes the outage probability of the information rate for AF relay channels. In this paper, we consider Rayleigh fading DF single relay channels with block Markov transmissions. Our focus is on the design of transmission strategies that optimize the outage probability or ergodic rate. In [9], a similar ergodic rate optimization problem was investigated. The key difference is that, at the transmitters of the source and relay, the strategies in [9] assume the availability of the complete knowledge of instantaneous channel state information (CSI), whereas the strategies in this paper only need the knowledge of variances of the channels (statistical CSI). Relative to the instantaneous CSI, the statistical CSI can be readily acquired at the trans- mitters and does not require frequent updates. We first derive the outage probability and erogdic rate by using a Hermitian quadratic form in complex Gaussian random variables (RVs). Based on the derived outage probability and ergodic rate, we next obtain the transmission strategies that either minimize the outage probability or maximize the ergodic rate, which we term as outage-optimal and ergodic- optimal strategies respectively. In deriving these transmission strategies, we restrict attention to two scenarios classified by the feasibility of spatial power allocation. In the scenario where no spatial power allocation is available, we show that the outage-optimal transmit signals, in contrast with the ergodic-optimal ones, are not necessarily independent, and their correlation generally depends on the SNRs of the links and the target transmission rate. Moreover, we further show that the ergodic rate is a monotonically decreasing function of the correlation coefficient between the transmit signals from the source and relay. In a small outage scenario, which is generally of practical interest, we derive an outage-optimal spatial power allocation strategy between source and relay. Our findings suggest that it is not always beneficial to use a relay, and the optimal power allocation generally depends on several parameters including the total transmit power, the target transmission rate, and the variances of the channels. Additionally, we present ergodic-optimal spatial power allo- cation strategies, which are obtained by numerically solving transcendental equations. The remainder of the paper is organized as follows. The system model of a single relay channel is introduced in Section II. The outage probability and the ergodic rate for Rayleigh fading DF relay channels are obtained in Section III. The 1536-1276/08$25.00 c  2008 IEEE