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Exact Performance Analysis of Dual-Hop Semi-Blind AF Relaying over Arbitrary Nakagami-m Fading Channels

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION Exact Performance Analysis of Dual-Hop Semi-Blind AF Relaying over Arbitrary Nakagami-m Fading Channels Minghua Xia, Chengwen Xing,
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION Exact Performance Analysis of Dual-Hop Semi-Blind AF Relaying over Arbitrary Nakagami-m Fading Channels Minghua Xia, Chengwen Xing, Yik-Chung Wu, and Sonia Aïssa, Senior Member, IEEE Abstract Relay transmission is promising for future wireless systems due to its significant cooperative diversity gain. The performance of dual-hop semi-blind amplify-and-forward AF relaying systems was extensively investigated, for transmissions over Rayleigh fading channels or Nakagami-m fading channels with integer fading parameter. For the general Nakagami-m fading with arbitrary m values, the exact closed-form system performance analysis is more challenging. In this paper, we explicitly derive the moment generation function MGF, probability density function PDF and moments of the endto-end signal-to-noise ratio SNR over arbitrary Nakagami-m fading channels with semi-blind AF relay. With these results, the system performance evaluation in terms of outage probability, average symbol error probability, ergodic capacity and diversity order, is conducted. The analysis developed in this paper applies to any semi-blind AF relaying systems with fixed relay gain, and two major strategies for computing the relay gain are compared in terms of system performance. All analytical results are corroborated by simulation results and they are shown to be efficient tools to evaluate system performance. Index Terms Amplify-and-forward AF, dual-hop, Meijer s G-function, Nakagami-m fading, performance analysis, semiblind relaying. I. INTRODUCTION RELAY transmission is promising for next-generation wireless systems, owing to its significant cooperative diversity gain []. A relay can be exploited to help the source forward signals to the destination if the direct channel from the source to the destination is in deep fading. In general, there are two kinds of relaying schemes, namely, decode-and-forward DF and amplify-and-forward AF. In contrast to DF relay, AF relay does not implement decoding operation and it simply amplifies the received signals prior to transmission; therefore, it is transparent to the modulation and coding schemes of the source. Manuscript received December, ; revised April 6, ; accepted June 8,. The associate editor coordinating the review of this paper and approving it for publication was A. Gulliver. M. Xia is with the Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology KAUST, Thuwal, Saudi Arabia C. Xing is with the School of Information and Electronics, Beijing Institute of Technology, Beijing, China Y.-C. Wu is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong S. Aïssa is with INRS-EMT, University of Quebec, Montreal, QC, Canada, and with KAUST, Thuwal, Saudi Arabia The work was supported in part by the HKU Seed Funding Programme, Project No , and by King Abdullah University of Science and Technology. Digital Object Identifier.9/TWC /$5. c IEEE In the AF technique, the relay gain aims to invert the firsthop channel while limiting the output power of the relay when the first-hop fading amplitude is low. Generally, there are three methods to determine the relay gain. The first method consists in fixing the relay gain regardless of the fading amplitude of the first-hop channel. Its performance degrades significantly if the channel amplitude varies a lot. The second method exploits full channel state information CSI. It requires a continuous estimate of the instantaneous first-hop channel amplitude, thus resulting in heavy training overhead. The third method, on the other hand, exploits only statistical CSI of the first-hop channel and it is called the semi-blind relay. Comparing with the full- CSI assisted relay, the semi-blind AF relay is more practical to be deployed, and it only introduces slight performance degradation [], [3]. The performance of dual-hop semi-blind AF relay has been extensively investigated, especially over Rayleigh fading channels [], [], [4] [7]. However, it is well-known that the Nakagami-m fading is more general and it can model extensive fading scenarios. For performance analysis of dualhop relay under Nakagami-m fading channels, in order to obtain closed-form performance metrics, it is usually assumed that both consecutive hops source-to-relay and relayto-destination experience fading with integer m values [3], [8] [4]. However, the propagation environments where the Nakagami fading parameter takes non-integer values are very common in practice, such as micro-cellular scenarios with strong specular components and land mobile satellite channels [5], [6]. Therefore, accurate performance evaluation of the relaying transmission over arbitrary Nakagami-m fading channels becomes extremely important for practical purpose. For the scenario with arbitrary m values, the lower bound on average symbol error probability SEP of full-csi assisted relaying transmission is studied in [7], by exploiting the conventional probability density function PDF based integration method. In [8], the moment generation function MGF-based approach is applied to analyze the average SEP, where the MGF is expressed by the Kampé deferiét s function. Due to its high complexity, the Kampé deferiét s function is hard to be further processed and, therefore, the average SEP is obtained by numerical integration [8]. In [9], a comprehensive framework for exact performance analysis over generalized fading channels is presented, including the Nakagaimi-m fading channels with integer m values, while for the Nakagami-m fading scenario with non-integer m, IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION performance bounds are obtained. To the best of the authors knowledge, exact and general performance analysis of dualhop relaying over Nakagami-m fading with arbitrary m values remains an open problem. This paper aims at filling this important gap. Specifically, dual-hop semi-blind AF relay over arbitrary Nakagami-m fading channels is considered in this paper. A general and compact MGF expression for the end-to-end SNR is first derived, by exploiting the generalized Meijer s Gfunction and Fox s H-function of two variables. The MGF is further processed and the PDF of the end-to-end SNR is explicitly developed. From the analytical MGF and PDF expressions, four important performance measures, namely, outage probability, average SEP, ergodic capacity and diversity order, are analytically obtained. In addition, the moments of the end-to-end SNR are derived and are further exploited to approximately evaluate the ergodic capacity. All analytical results developed in this paper are corroborated by simulation results, indicating the accuracy of the provided results. The rest of this paper is organized as follows. Section II describes the system model. The MGF, PDF and moments of the end-to-end SNR are derived in Section III. Four system performance measures, including outage probability, average SEP, ergodic capacity and diversity order, are investigated in Section IV. Numerical results are presented in Section V. Finally, Section VI draws the conclusion. II. SYSTEM MODEL We consider a dual-hop relaying system with a single AF relay. The source, relay, and destination are equipped with single half-duplex antenna. Both the first hop source-to-relay and the second hop relay-to-destination experience independent but not necessarily identically distributed Nakagami-m fading with fading shape factors m,m.5, respectively. The transmission period from the source to the destination is divided into two consecutive phases. During the first transmission phase, the source S transmits signal x with energy to the relay R. Accordingly, the received signal at the relay is given by y SR = h SR x + n SR, where h SR denotes the complex channel gain between the source and the relay, and n SR is the additive white Gaussian noise AWGN at the relay with zero mean and variance σr. During the second phase, the source keeps silent while the relay amplifies its received signal y SR and forwards it to the destination D. Consequently, the received signal at the destination is y RD = βh RD h SR x + n SR +n RD = βh RD h SR x + βh RD n SR + n RD, where β stands for the relay gain, h RD denotes the complex channel gain between the relay and the destination, and n RD is the AWGN at the destination with zero mean and variance σd. According to, the end-to-end SNR from the source to the destination is given by [] = βh RDh SR βh RD σ R + σ D = + c, 3 where h SR /σr and h RD /σd refer to the instantaneous SNRs at the first hop and at the second hop, respectively, and c is a constant for a fixed relay gain β: c β σr. 4 III. THE MGF, PDF AND MOMENTS OF THE END-TO-END SNR In this section, the exact closed-form MGF, PDF and moments of the end-to-end SNR under arbitrary Nakagami- m fading channels are derived. A. MGF of the End-to-End SNR With the assumption of Nakagami-m fading channels, the received SNR i at the i th hop, where i =,, is of Gamma distribution given by [, Eq.. ]: f i i = mmi i mi exp m i Γm i mi i i, 5 i i }{{} β II = A σ R A i where Γ. is the Gamma function, E{ h SR } /σr = m m + /σr, and E{ h RD } /σd = m m + /σd, with E{. } denoting the statistical expectation operator. For the semi-blind AF relay transmission, the relay gain is determined by the channel statistics at the first hop. In general, there are two major strategies to calculate the relay gain: Strategy I: The square of relay gain is given by [] βi E = S E E { h SR } + σr = S σr +. 6 Strategy II: The square of relay gain is given by [] { } βii = E h SR + σr = E { } S σr E. 7 + Substituting 5 into 7 yields m m d = A Γm σ R + exp Ψ m,m ; m, 8 where we exploited [, vol., Eq ] in 8 with Ψa, b; z being the Tricomi confluent hypergeometric function [, vol. 5, p. 593]. From 6 and 8, it is clear that once, σr and m are known, βi and β II are fixed. Since the end-to-end SNR expression in 3 depends only on the relay gain β, through c in 4, the following analysis holds for any semi-blind AF relaying systems with fixed relay gain. Therefore, in the following, we do not distinguish the analysis for the two strategies above, and their system performance and comparison will be examined in detail in Section V. With a fixed relay gain β, c in 4 is fixed and, thus, we can calculate the MGF of the end-to-end SNR in 3. This is summarized in the following theorem. Theorem : The MGF of the end-to-end SNR of dual-hop semi-blind AF relaying systems over arbitrary Nakagami-m XIA et al.: EXACT PERFORMANCE ANALYSIS OF DUAL-HOP SEMI-BLIND AF RELAYING OVER ARBITRARY NAKAGAMI-M FADING CHANNELS 3 M s = Γm Γ m Γm,,,, G, [: ],, [: ] m + s cm m k k k= k + m m ; + m. 9 ; fading channels is given by 9 at the top of this page, where. stands for the integer ceiling operator and G[.. ] denotes the generalized Meijer s G-function of two variables defined in 3 of Appendix A. Proof: See Appendix B. Although the G-function in 9 cannot be directly calculated by popular mathematical softwares such as Matlab and Mathematica, it can be easily evaluated by the algorithm recently developed in [5, Table II], which is based on the double Mellin-Barnes type integrals. On the other hand, in order to show more extensive applications of Theorem, we will derive an analytical expansion for the MGF in the next subsection. Special Case: When m takes integer values, that is, m =, the generalized G-function of two variables in 9 reduces to the conventional Meijer s G-function. Specifically, we have the following proposition. Proposition : For b =and x =, the generalized Gfunction of two variables reduces to the conventional Meijer s G-function of one variable, that is,,,,, G, [: ],, [: ] a x b ; b y = G,, ; x. a, b Proof: See Appendix C. Based on Proposition, it is straightforward to show that when m takes integer values, 9 reduces to M s = m k m Γm Γm k k= cm m, m + s k + m,m G,, which is equivalent to the result reported in [9, Eq. 8]. B. PDF of the End-to-End SNR Although the MGF is explicitly given in 9, the PDF cannot be directly obtained by applying the inverse Laplace transform due to the high complexity of the involved G-function. In order to proceed, we derive an analytical expansion for the MGF. More specifically, from the expression in 3 of Appendix A and by using the residue theorem, this G-function can be equivalently evaluated by considering the residues at the poles of the integrand in 3 [3, p. 538]. Therefore, the MGF in 9 can be reformulated using [3, Eq..3], thus yielding k M s = m k k k= k + m p+q m p m q p! q! p= q= m p + s q, cm m where a k = Γa + k/γa is the Pochhammer symbol. Then, exploiting the symbolic operators proposed in [7], it can be shown that the double series in can be transformed into a single series of the product of two generalized hypergeometric functions and, thus, the MGF can be reformulated as 3-4 at the top of next page, where we exploited the identity F a, b; ; /x = x a Ψa; +a b; x [8, Eq. 3.. ] in 4. Finally, performing the inverse Laplace transform of 4 leads to the PDF of the end-to-end SNR as summarized in the following corollary, where we exploited [, vol. 5, Eq ] as well as the scaling and shifting properties of the inverse Laplace transform. Corollary : The PDF of the end-to-end SNR of dualhop semi-blind AF relaying systems over arbitrary Nakagami- m fading channels is given by 5 in the middle of next page, where p F q. stands for the generalized hypergeometric function [6, Eq. 4.. ]. C. Moments of the End-to-End SNR Although the moments of the received SNR can be obtained by taking the derivatives of the MGF in 9 and setting s =, the resultant expressions are extremely complicated. On the other hand, the moments with a simple form can be calculated in a straightforward manner and they are summarized in the following lemma. Lemma : The n th -order moment of the end-to-end SNR of dual-hop semi-blind AF relaying systems is given by n m E{ n cm } = m n m n m Ψ m + n, m +; cm. 6 Proof: See Appendix D. For the ease of subsequent discussion, the first-order and second-order moments of the end-to-end SNR are explicitly expressed as follows. Corollary : The first-order and second-order moments of the end-to-end SNR of dual-hop semi-blind AF relaying 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION M s = = k m k k r! k + m r m r m r k= r= r F k + m + r, m + r; ; r m + s F k + m + r,m + r; ; m + s cm m cm m m k+m m k k r! k + m r m r m r k= r= m r cm F k + m + r, m + r; ; m + s Ψ k+m k + m + r, +k m + m ; cm m m + s 3. 4 f = k= k cm exp m k+m m k r= r! k + m r m r m r m r F k + m + r, m + r; ; m [ Γ k + m m k+m Γr + m Γk + m F k + m + r; +k m + m,k+ m ; cm m + Γk m k+m m + m cm m Γk + m + rγm F r + m ; k + m m,m ; cm m m ]. 5 systems are given by and m cm E{} = m Ψ m +,m +; cm 7 E{ } = m m m +m + cm m Ψ m +,m +; cm, 8 respectively. Proof: The results can be proved by substituting n = and n =into 6, and with straightforward manipulations. IV. FURTHER PERFORMANCE ANALYSES After obtaining the MGF, PDF and moments of the end-toend SNR, we investigate in this section the outage probability, average SEP, ergodic capacity and diversity order of the dualhop semi-blind AF relay transmission. A. Outage Probability The outage probability is defined as the probability that the instantaneous SNR falls below a predefined threshold value th. Mathematically, the outage probability is expressed as P outage th =Pr{ th } = th f d. 9 In order to obtain a closed-form expression of 9, we first rewrite the PDF in 5 by applying the infinite series expansion of the exponential function [, vol., Eq ]. Then, substituting the formulation of 5 into 9 and making use of [, vol. 3, Eq.... ], after some manipulations, we finally obtain at the top of next page, where Ba, b is the Beta function. Although it seems complicated, mainly involves common Gamma function, Beta function and generalized hypergeometric function. Hence, is easy to be numerically evaluated and its accuracy is corroborated by the simulation results in Section V. In [9], an approximate analysis for the performance of dual-hop AF relaying over Nakagami-m fading channels with non-integer m values is presented, based on an upper bound XIA et al.: EXACT PERFORMANCE ANALYSIS OF DUAL-HOP SEMI-BLIND AF RELAYING OVER ARBITRARY NAKAGAMI-M FADING CHANNELS 5 P outage th = m k+m m k k r! k + m r m r m r k= r= m r cm F k + m + r, m + r; ; cm [ m p p+k+m Γ k + m m th Bp + k + m, p! Γr + m p= Γk + m F 3 k + m + r,p+ k + m ;+k m + m,k+ m,p+ k + m +; cm m th + Γk m k+m m + m cm m p+m th Bp + m, Γk + m + rγm F 3 r + m,p+ m ; k + m m,m,p+ m +; cm ] m th. on the end-to-end SNR [9, Eq. 5]: min {, c }. For the purpose of comparison, the lower bound on the outage probability is reproduced as follows [9, Eq. 8]: P L outage = th Γm m, m th + Γm Γm m+k cm m k= Γ k m + kk! m m k, m th th, where a, x = x ta e t dt denotes the lower incomplete Gamma function and Γa, x = x ta e t dt stands for the upper incomplete Gamma function. The accuracy of compared to will be illustrated in Figs. -3, and is detailed in Section V. B. Average Symbol Error Probability By exploiting the MGF in 9, it is straightforward to obtain the average SEP of different kinds of modulation schemes [9], [3]. Herein, we focus on the M-PSK constellation since it is adopted in 3GPP long-term evolution LTE systems [3]. For the coherently detected M-PSK, the average SEP is given by [, Eq. 8. 3] P s E = Θ gpsk M π sin dθ, 3 θ where the constants g PSK = sin π/m and Θ = M π/m. Although its closed-form expression cannot be obtained, 3 can be easily evaluated numerically since it is a proper integral with finite limits and closed-form integrand. On the other hand, a closed-form approximation of the average SEP can be obtained by substituting the MGF into the following expression [3, Eq. ] Θ P s E π M g 6 PSK M 3 g PSK Θ + π gpsk M 4 sin. 4 Θ For the numerical evaluation of 3 and 4, 4 is exploited, which mainly involves the generalized hypergeometric function and can be easily calculated. The accuracy of 3 and 4 will be demonstrated in Section V. C. Ergodic Capacity The ergodic capacity is defined as the statistical mean of the instantaneous mutual information between the source and the destination, in the unit of bit/s/hz. Mathematically, C erg = log + f d, 5 which can be numerically evaluated by exploiting the analytical expression of f in 5. The factor / is introduced by the fact that two transmission phases are involved. On the other hand, in view of the moments of the end-to-end SNR in 6, a closed-form expression of the ergodic capacity can be obtained by exploiting the Taylor series expansion of log + with respect to +E{}. More specifically, C erg = E{ln + } ln = ln + E{} ln { + } ln E E{} n ln n + E{} n! n= = ln ln+e{} E{ } E {} 4 ln + E{} + o [ E{} 3], 6 where o[.] denotes the Landau notation and E{} and E{ } are explicitly shown in 7 and 8, respectively. The accuracy of the numerical integration 5 and its approximation 6 will be examined in Section V, in comparison with the simulation results. Here it is worth mentioning that a unified MGF-based approach for computing the ergodic capacity over generalized 6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION fading channels was recently developed in [33, Eq. 7]. Unfortunately, the unified approach cannot be applied here, due to the complicated structure of the MGF in 4, and it is expected that no closed-form capacity expression can be derived in terms of conventional special functions. On the other hand, notice that, although the Taylor formula is also exploited in [9, Eq. 6], the ergodic capacity analysis there cannot be applied in the general Nakagami-m fading case with non-integer m values, given that the finite-sum expansion of the incomplete Gamma function was exploited in [
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