Closed-Form Bounds for Multihop Relayed Communications in Nakagami-m Fading George K. Karagiannidis ∗ , Theodoros A. Tsiftsis † , Ranjan K. Mallik ‡ , Nikos C. Sagias § , and Stavros A. Kotsopoulos † ∗ Department of Electrical & Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece, Email: geokarag@auth.gr † Department of Electrical & Computer Engineering, University of Patras, Rion, 26500 Patras, Greece, Email: {tsiftsis;kotsop}@ee.upatras.gr ‡ Department of Electrical Engineering, Indian Institute of Technology, Hauz Khas, 110016 New Delhi, India, Email: rkmallik@ee.iitd.ernet.in § Laboratory of Electronics, Physics Department, University of Athens, Panepistimiopolis, Zografou, 15784 Athens, Greece, Email: nsagias@space.noa.gr Abstract—We present closed-form bounds for the performance of multihop transmissions with non-regenerative relays over Nakagami-m fading channels. The end-to-end signal-to-noise ratio (SNR) is formulated and upper bounded by using the inequality between harmonic and geometric mean of positive random variables (RVs). Novel closed-form expressions are deri- ved for the moment generating function, the probability density function, and the cumulative distribution function of the product of arbitrary powers of statistically independent Gamma RVs in terms of the Meijer’s G-function. Using these theoretical results, closed-form lower bounds are obtained for the outage and average bit error probability of phase or frequency modulated signallings, while simple asymptotic expressions are also given for the bounds at high SNRs. Numerical results are compared to computer simulations, to show the tightness of the proposed bounds. I. I NTRODUCTION Multihop systems realize a number of advantages over traditional communications systems in the areas of deploy- ment, connectivity and capacity while minimize the need for fixed infrastructure. Relaying techniques enable network connectivity where traditional architectures are impractical due to location constraints and can be applied to cellular, wireless local area networks (WLAN), and hybrid networks. In multihop systems the source-terminal communicates with the destination-terminal through a number of relays-terminals, having the advantage of broadening the coverage without using large transmitting power [1]–[5]. The concept of cooperative diversity, where the mobile users cooperate each other in order to exploit the benefits of spatial diversity without the need of using physical antenna arrays, has also gained great interest [6]–[9]. The performance analysis of multihop wireless commu- nication systems operating in fading channels has been an important field of research in the past few years. Hasna and Alouini have presented a useful and semi-analytical framework for the evaluation of the end-to-end outage probability of multihop wireless systems with non-regenerative channel state information (CSI)-assisted relays over Nakagami-m fading channels [3]. Moreover, the same authors have studied the dual-hop systems with regenerative and non-regenerative (CSI- assisted or fixed gain) relays over Rayleigh [1], [4] and Nakagami-m [2] fading channels. Recently, Boyer et al. [5], have proposed and characterized four channel models for multihop wireless communication and also have introduced the concept of multihop diversity. Finally, Karagiannidis et al. have studied the performance bounds for multihop wireless communications with blind (fixed gain) relays over Rice, Hoyt and Nakagami-m 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-m fading. In this paper, using the well-known inequality between harmonic and geometric means of positive random variables (RVs), we present efficient performance bounds for the end-to- end signal-to-noise ratio (SNR) of multihop wireless commu- nication systems with CSI-assisted or fixed gain relays opera- ting in non-identical Nakagami-m fading channels. Motivated by the fact that the proposed bounds, in their general form, are products of arbitrary powers of statistically independent squared Nakagami-m (Gamma) RVs, we derive novel closed- form expressions for their moment generating function (MGF), probability density function (PDF), and cumulative distribution function (CDF) in terms of the Meijer’s G-function. Using these expressions, closed-form lower bounds are presented for important end-to-end system performance metrics, such as outage probability and average bit error probability (ABEP) for binary phase shift-keying (BPSK) and binary frequency shift- keying (BFSK) modulation schemes, while simple asymptotic expressions are also given for the bounds at high SNRs. Nu- merical and computer simulation examples verify the accuracy of the presented mathematical analysis and show the tightness of the proposed bounds. II. STATISTICAL BACKGROUND Theorem 1: (MGF of the product of arbitrary powers of Gamma RVs): Let {X i } N i=1 be N independent, but not neces- sarily identically distributed (i.n.i.d.), Gamma RVs, with PDF given by f Xi (x)= x αi-1 β αi i Γ(α i ) exp  - x β i  (1)