IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 9, SEPTEMBER 2009 2551 A Unified Framework for Performance Analysis of CSI–Assisted Cooperative Communications over Fading Channels Marco Di Renzo, Associate Member, IEEE, Fabio Graziosi, Member, IEEE, and Fortunato Santucci, Senior Member, IEEE Abstract—In this Letter, we propose a comprehensive frame- work for performance analysis of cooperative wireless systems using Amplify and Forward (AF) relay methods. The frame- work relies on the Moment Generating Function (MGF–) based approach for performance analysis of communication systems over fading channels, and on some properties of the Laplace Transform, which allow to develop a single–integral relation between the MGF of a random variable and the MGF of its inverse. Moreover, a simple lower bound for Outage Probability (Pout ) and Outage Capacity (OC) computation is also introduced. Numerical and simulation results are provided to substantiate the accuracy of the proposed framework. Index Terms—Cooperative diversity, multi–hop networks, gen- eralized fading channels, performance analysis. I. I NTRODUCTION D IVERSITY combining is a well–known concept widely used in many current wireless technologies to improve the overall system performance by mitigating the signal fad- ing arising from multipath propagation due to the wireless medium [1], [2]. Recently, a new concept of spatial diversity has received significant attention in the research community: cooperative diversity [3]–[5]. In these systems each mobile radio becomes part of a large distributed array, and shares its single–antenna to assist the communication between two neighboring source and destination radios by using relayed transmissions, and distributed diversity combining techniques [6], [7]. In this contribution, we propose a comprehensive frame- work for performance analysis of a generic cooperative sys- tem, often denoted as multi–hop multi–branch network. The framework is intended to overcome the limitations that can be encountered in the current technical literature available so far, e.g., [8]–[15]. In particular, i) most analytical frameworks con- sider only the dual–hop scenario, and in most circumstances diversity is not considered (see, e.g., [10], [11]), ii) some analytical developments used for the analysis of dual–hop Paper approved by G. K. Karagiannidis, the Editor for Fading Channels and Diversity of the IEEE Communications Society. Manuscript received December 20, 2007; revised April 2, 2008, June 20, 2008, and August 19, 2008. This paper was presented in part at the IEEE International Conference on Communications (ICC), Beijing, China, May 2008. M. Di Renzo is with The University of Edinburgh, College of Science and Engineering, School of Engineering, Institute for Digital Communications (IDCOM), Alexander Graham Bell Building, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JL, Scotland, United Kingdom (UK) (e-mail: m.di- renzo@ed.ac.uk). The research activity of Marco Di Renzo was supported, in part, by the Torres Quevedo 2008 Program’s aid (PTQ–08–01–06437), and the research projects PERSEO (TEC2006–10459/TCM), LOOP (FIT–330215– 2007–8), and m:VIA (TSI–020301–2008–3). F. Graziosi and F. Santucci are with the Department of Electrical and Information Engineering and the Center of Excellence in Research DEWS, University of L’Aquila, 67040 Poggio di Roio, L’Aquila, Italy (e-mail: {fabio.graziosi, fortunato.santucci}@univaq.it). Digital Object Identifier 10.1109/TCOMM.2009.09.070653 systems do not seem to be easily generalizable to the multi– hop scenario (see, e.g., [10], [11]), iii) the vast majority of the developed frameworks are either specific to some channel models or in some cases they loose accuracy in the high Signal–to–Noise Ratio (SNR) region, and for non–identically distributed hops and diversity branches [13], [14], and iv) almost always, only multipath fading is considered in the anal- ysis, while the more realistic composite multipath/shadowing channel model has not been addressed so far. More specifically, we consider Channel State Information (CSI–) assisted Amplify and Forward (AF) relay methods. The framework relies on the Moment Generating Function (MGF) based approach for performance analysis of communication systems over fading channels [2], and on some properties of the Laplace Transform, which allow to develop a simple single–integral relation between the MGF of a random variable and the MGF of its inverse. By exploiting this integral relation, we develop explicit formulas to estimate Average Bit Error Probability (ABEP), Outage Probability (P out ), average SNR, and Outage Capacity (OC), which are applicable to any environment with arbitrary fading distribution. For the sake of completeness, we observe that a similar approach was suggested in [16], which, however, foresaw the computation of two–fold numerical integrations. In this contribution, by using some properties of the Laplace Transform, we develop an al- ternative integral representation that requires the evaluation of only a single finite–limit integral, thus allowing the approach to be used for any typical channel model encountered in practice. Finally, we also propose a lower bound for efficiently and quickly computing P out and OC over generalized fading channels. This lower bound is intended to provide an efficient and simple tool for performance assessment and comparison of various system settings, without the need to perform either extensive simulations or numerical computations. This bound shows two main characteristics: i) with respect to [9], it accounts for spatial diversity as well, and reduces to the latter framework when spatial diversity is not considered, and ii) with respect to [13], it provides exact results in the absence of spatial diversity (i.e., multi–hop networks). The remainder of the Letter is organized as follows. Section II describes the system model. In Section III, the proposed framework is developed, and specific results for various fading distributions are provided. In Section IV, the lower bound for P out and OC computation is introduced. Finally, Section V shows some numerical results to validate the proposed framework, and Section VI concludes the Letter. II. SYSTEM MODEL Let us consider (see Fig. 1) a multi–branch multi–hop coop- erative network with  virtual diversity branches and   relays 0090-6778/09$25.00 c ⃝ 2009 IEEE