Dual-Hop AF Relaying Systems in Mixed Nakagami-m and Rician Links Samy S. Soliman, Student member, IEEE, and Norman C. Beaulieu, Fellow, IEEE AITF Wireless Communications Laboratory University of Alberta, Edmonton, Alberta, T6G2V4 Canada {soliman, beaulieu}@icoremail.ece.ualberta.ca Abstract—New, exact closed-form expressions for the prob- ability density function (PDF) and the cumulative distribution function (CDF) are derived for the instantaneous received end-to- end signal-to-noise ratio (SNR) of dual-hop amplify-and-forward (AF) relaying systems operating over mixed Nakagami-m and Rician fading links. The expressions are used to obtain exact integral solutions for the ergodic capacity and the average symbol error probability as well as an exact closed-form solution for the outage probability of dual-hop AF systems operating over mixed links. The results obtained represent the first exact results for the cases of composite Nakagami-m/Rician fading links. The exact performance metrics are compared to performance bounds in the literature, and it is shown that the performance bounds are not tight for medium ranges of SNR. The effects of fading parameters on the system performance are studied. It is shown that the limiting slopes of the average error probability and outage probability curves are not affected by the fading parameters, however, SNR gains are achieved by increasing the fading parameter. Moreover, it is shown also that an increase in the Rician parameter, K, results in a notable SNR gain in the system performance, while an increase in the Nakagami- m parameter, m, has diminishing returns and gives negligible improvement in the system performance in some instances. Index Terms—Amplify-and-forward, average symbol error probability, cooperative networks, dual-hop relaying, ergodic capacity, mixed links, outage probability. I. I NTRODUCTION Wireless cooperative communication has become an impor- tant subject for study because it promises an advancement in the quantity as well as the quality of services provided to users [1]–[10]. In [1], the authors presented an approximation for the end-to-end received signal-to-noise ratio (SNR) using the harmonic mean of two independent exponential random variables in order to estimate the average error probability and outage probability of dual-hop AF relaying systems operating over Rayleigh fading links. In [2], the work was extended to the case of multihop transmission over Nakagami-m fading channels, and the authors obtained an approximation to the system outage probability. Since then, many publications, such as [3], [4] have used the direct harmonic mean approxima- tion of the individual per hop instantaneous SNRs to obtain performance bounds. However, it has been shown that such bounds are neither tight for small-to-moderate values of SNR, nor for Nakagami-m fading channels for large values of m [5], [7]. In [5], an approximation has been proposed that has the same computational complexity as previous bounds in the literature, while being more accurate, especially for small-to- medium values of SNR. The new approximation gives more accurate prediction of the exact performance than the bounds. In [6], [7], a new analytical approach, the generalized transformed characteristic function (GTCF) method, was pro- posed to obtain the first exact results published for the outage probability, ergodic capacity and the average symbol error probability of multihop AF relaying systems operating over general fading channels. An important result shown in [6], [7] is that as the number of hops increases, the ergodic capacity of the system decreases although other performance metrics of the systems are enhanced. In the following, we study dual-hop AF networks and obtain exact solutions for the performance metrics of dual-hop AF systems operating over mixed Nakagami-m and Rician fading channels [8]–[10]. Previously, the authors in [8] obtained an exact expression for the outage probability of dual-hop AF systems operating over mixed Rayleigh and Rician fading channels; however, that exact expression was not used to obtain other performance metrics. Instead, the authors used an approximation of the end-to-end SNR to obtain the average error probability of the system. The authors also presented a lower bound to the average error probability based on approximating the end-to-end SNR by the minimum value of the individual links fading SNRs. In [9], the authors studied also the case of mixed Rayleigh and Rician fading chan- nels for dual-hop, fixed gain relaying systems, and obtained infinite-series representations for the outage probability and the average bit error probability of these systems. The case of dual-hop AF systems with mixed Nakagami-m and Rician fading links was studied in [10]. The authors in [10] also used the approximation used in [8], due to its mathematical tractability, to approximate the outage probability and the average error probability of the proposed system. Note that the results presented in [8] and [10] are based on an approximation for the end-to-end SNR and are not exact. In this paper, we derive exact closed-form expressions for the PDF and the CDF of the end-to-end SNR of dual- hop, variable gain AF relaying systems operating over mixed Nakagami-m and Rician fading channels and we obtain the first exact analytical results for the average error probability, outage probability and ergodic capacity of such dual-hop AF systems. The most important differences between the work presented here and previous literature are: 1) The closed-form GC'12 Workshop: The 7th IEEE International Workshop on Heterogeneous, Multi-Hop, Wireless and Mobile Networks 978-1-4673-4941-3/12/$31.00 ©2012 IEEE 447