18 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 1, JANUARY 2006 Bounds for Multihop Relayed Communications in Nakagami- Fading George K. Karagiannidis, Senior Member, IEEE, Theodoros A. Tsiftsis, Member, IEEE, and Ranjan K. Mallik, Senior Member, IEEE Abstract—We present closed-form lower bounds for the perfor- mance of multihop transmissions with nonregenerative relays over not necessarily identically distributed Nakagami- fading chan- nels. The end-to-end signal-to-noise ratio is formulated and upper bounded by using an inequality between harmonic and geometric means of positive random variables (RVs). Novel closed-form expressions are derived for the moment generating function, the probability density function, and the cumulative distribution func- tion of the product of rational powers of statistically independent Gamma RVs. These statistical results are then applied to studying the outage probability and the average bit-error probability for phase- and frequency-modulated signaling. Numerical examples compare analytical and simulation results, verifying the tightness of the proposed bounds. Index Terms—Average bit-error probability (ABEP), Gamma random variables (RVs), multihop relayed communications, Nak- agami- fading, outage probability. I. INTRODUCTION M ULTIHOP systems have a number of advantages over traditional communication networks in the areas of deployment, connectivity, and capacity, while minimizing the need for fixed infrastructure. Relaying techniques enable network connectivity where traditional architectures are im- practical due to location constraints, and can be applied to cellular, wireless local area networks (WLANs), and hybrid networks. In multihop systems, the source terminal communi- cates with the destination terminal through a number of relay terminals. Therefore, multihop systems have the advantage of broadening the coverage without using large transmitting power [1]–[5]. Recently, the concept of cooperative diversity, where the mobile users cooperate/collaborate with each other in order to exploit the benefits of spatial diversity without the need for using physical antenna arrays, has gained great interest. In general, cooperative networks are multihop communication networks, where the destination terminal combines the signals received from both the source terminal and relays [6]–[9]. The performance analysis of multihop wireless commu- nication systems operating in fading channels has been an Paper approved by M.-S. Alouini, the Editor for Modulation and Diversity Systems of the IEEE Communications Society. Manuscript received August 21, 2004; revised January 20, 2005; April 19, 2005; and June 24, 2005. This paper was presented in part at the IEEE International Conference on Communications, Seoul, Korea, May 2005. G. K. Karagiannidis is with the Department of Electrical and Computer Engi- neering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: geokarag@auth.gr). T. A. Tsiftsis is with the Department of Electrical and Computer Engineering, University of Patras, Rion, GR-26500 Patras, Greece (e-mail: tsiftsis@ee.upa- tras.gr). R. K. Mallik is with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, New Delhi 110016, India (e-mail: rk- mallik@ee.iitd.ernet.in). Digital Object Identifier 10.1109/TCOMM.2005.861679 important field of research in the past few years. Hasna and Alouini have presented a useful and semianalytical framework for the evaluation of the end-to-end outage probability of multihop wireless systems with nonregenerative channel state information (CSI)-assisted relays over Nakagami- fading channels [3]. Moreover, the same authors have studied the outage and the error performance of dual-hop systems with regenerative and nonregenerative (CSI-assisted or fixed-gain) relays over Rayleigh [1], [4] and Nakagami- [2] fading chan- nels. Recently, Boyer et al. [5] have proposed and characterized four channel models for multihop wireless communications, and have also introduced the concept of multihop diversity. Finally, Karagiannidis has studied the performance bounds for multihop relayed transmissions with blind (fixed-gain) relays over Nakagami- (Rice), Nakagami- (Hoyt), and Nakagami- fading channels [10] using the moments-based approach [11]. However, to the best of the authors’ knowledge, the performance of multihop relayed systems has never been addressed in terms of tabulated functions in Nakagami- fading. In this letter, using the well-known inequality between har- monic and geometric means of positive random variables (RVs), we present performance bounds for the end-to-end signal-to- noise ratio (SNR) of multihop wireless communication systems with CSI-assisted or fixed-gain relays operating in nonidentical Nakagami- fading channels. Motivated by the fact that the proposed bounds, in their general form, are products of rational powers of statistically independent squared Nakagami- RVs (or equivalently, Gamma RVs), we derive novel closed-form expressions for their moment generating function (MGF), the probability density function (PDF), and the cumulative distri- bution function (CDF). These statistical results are then applied to the study of important system performance metrics. Closed- form lower bounds are derived for the outage probability, and the average bit-error probability (ABEP) for binary phase-shift keying (BPSK) and binary frequency-shift keying (BFSK) mod- ulation schemes. Numerical and computer simulation examples verify the accuracy of the presented mathematical analysis and show the tightness of the proposed bounds. The remainder of this letter is organized as follows. In Sec- tion II, closed-form expressions for the MGF, PDF, and CDF of the product of rational powers of Gamma RVs are presented. Next, Section III introduces the multihop system and channel model under consideration. In the same section, closed-form upper bound expressions for the statistics of the end-to-end SNR are proposed both for CSI-assisted and fixed-gain relayed sys- tems. These results are applied in Section IV to evaluate the end-to-end performance metrics of multihop wireless commu- nication systems. Finally, some concluding remarks are given in Section V. 0090-6778/$20.00 © 2006 IEEE