Performance Analysis of Dual-Hop systems with Fixed-Gain Relays over Generalized 17-J-L Fading Channels

Transcription

1 Globecom Wireless Communications Symposium Performance Analysis of Dual-Hop systems ith Fixed-Gain Relays over Generalized 17-J-L Fading Channels Osamah S. Badarneh and Michel Kadoch Abstract-The performance of dual-hop ireless communication systems ith semi-blind amplify-and-forard (A F) relays over independent, but not necessarily identically distributed, 1]-/1> fading channels is investigated. This performance is studied based on the evaluation of the outage probability, average bit-error probability (ABEP), and average end-to-end received signal-to-noise ratio (SNR). To this end, novel expressions for the outage probability and average end-to-end received SNR are derived. Moreover, an expression for end-to-end SNR of the probability density func. tion (PDF) is derived. Based on the derived PDF, e then obtam a novel expression for the generalized moments of the received SNR.. We conduct extensive simulation in order to study the effect of the fading parameters (i.e., 1] and /1» on the system performance. Our findings sho that, in contrast to the fading parameter. "1' /1> has a significant impact on the system performance. In addition, simulation results demonstrate that, the derived expressions match very ell ith the numerical simulations. I. INTRODUCTION Dual-hop transmission can be exploited hen the direct link beteen the source and destination terminals is deeply faded. In such a sy stem the transmission is accomplished ith the help of a relay terminal. The source terminal transmits the signal to the relay terminal, and then the relay terminal retransmits the received signal to the destination. In dual-hop transmission systems, relays can be classified into to main categories, namely, amplify-and forard (AF) and decode-and-forard (DF). In the former, the relay just amplifies and forards the received signal ithout perfonning any sort of decoding. In the latter, the relay fully decodes the incoming signal and then retransmits the decoded version to the destination. Recently, dual-hop transmission systems employing AF relaying have been studied over different fading channel models [1]-[7]. In [l], the end-to-end performance of a dual-hop communication system ith both AF and DF relaying and operating over flat Rayleigh fading channels is evaluated. For both relay types, the authors derived c1osed-fonn expressions for the outage probability. In addition, the considered system is studied under the effect of the relay saturation. Costa and Yacoub investigated the performance of dual-hop communication systems equipped ith semi-blind relays over Nakagami-m fading channels [2]. They evaluated the performance of their system based on the end-to-end signal to noise ratio (SNR). The end-to-end performance of a dual-hop ireless communication system ith a fixed-gain relay over K fading channels as investigated in [3]. The authors evaluated the considered system in terms of the average bit error probability (ABEP) assuming BPSK, amount of fading (AoF), and end-to-end SNR. The authors in [4] studied the performance of dual-hop ireless communication systems operating over independent, but not necessarily identically distributed, generalized garmna fading channels. They provided novel expressions for the end-to-end SNR, AoF, and average symbol error probability (ASEP) for different digital modulation techniques. In [5], approximated closed-form expressions for the outage probability, ABEP, and average channel capacity are derived and evaluated. In [6], O. Badarneh is ith the Department of Electrical Engineering at University of Tabuk, KSA. M. Kadoch is ith the Department of Electrical Engineering at Ecole de Technologie Superieure, Canada /12/$ IEEE the authors investigated the system performance of AF dualhop relay system over mixed fading distributions, namely, Nakagami-m and Rician. The system performance is studied in tenns of outage probability and ASEP. The authors in [7] derived expressions for the moment generation function (MGF), probability density function (PDF), and moments of the end-to-end SNR over arbitrary Nakagami-m fading channels ith semi-blind AF relay. Based on these results, the authors evaluated their system in terms of the outage probability, ASEP, and ergodic capacity. In literature, there exists a large number of distributions that ell describe the ireless channels, such as Rayleigh, Rice, Nakagami-m, and Hoyt. Hoever, in [8], the authors introduced a ne fading channel model, namely the 'f/ - J.L model. This model is more flexible than the other fading models and can yield better fits to experiential data. The 'f/ - J.L model includes Nakagami-m and Nakagami-q models as special cases. In this paper, e focus on semi-blind AF dual-hop systems and analyze their end-to-end performance over independent, but not necessarily identically distributed, 'f/ - J.L fading channels. The main contribution of this paper is to derive expressions for the average end-to-end SNR, the end-to-end outage probability, and the moment generating (MGF) of the end-to-end SNR of this dual hop system. In the course of our analysis, e derive novel expressions for end-to-end SNR of the cumulative distribution function (CDF), and the probability density function (PDF). Also, an expression for the parameter C hich describes the relay gain is derived. With these results, the performance of the considered system is then evaluated under different scenarios, such as varying average SNR per hop, fading parameters, and poer imbalance beteen the to hops. It is orth mentioning here that the derived expressions can be reduced to previously knon results, such as Nakagami-m and Rayleigh fading distributions [1], [2]. The remainder of the paper is organized as follos. Section II describes the system and channel models. The MGF, CDF, PDF, and moments of the end-to-end SNR are derived in Section III. Numerical and simulation results are presented in Section IV. Finally, Section V dras the conclusion. II. SY STEM AND CHANNEL MODELS A. System Model Consider a non-regenerative dual-hop ireless communication system ith a single semi-blind AF relay as shon in Figure 1. In such a system, the communication beteen a source terminal (S) and a destination terminal (D) is facilitated by a relaying terminal (R), hich amplifies the received signal ithout any sort of decoding and then transmits it to the destination terminal. Non-regenerative systems ith semi-blind (or blind) relays introduce fixed gain on the received signal regardless of the amplitude fading of the first hop (i.e., S -+ R). We assume that the source, relay, and destination terminals are equipped ith single half-duplex antenna. In addition, the fading amplitude of the j-th hop (O;j, here (j = 1,2) is assumed to be an 'f/ - J.L random variable. Assuming that the source S transmits a signal ith an average poer nonnalized to unity, then the end-to-end SNR 4148

2 III. PERFORMANCE ANALY SIS Fig. 1. is given in [1] as: _. -> Wireless link 0 Relaying terminal o Source terminal 0 Destination terminal Dual-hop ireless system ith semi-blind relay. "fend = (ai/ No1) (a / No2) (a /No2) + (1/C 2 NoJ here No1, and N02 are the single-sided poer spectral density of the additive hite Gaussian noise (AWGN) at the first and second hops, respectively. C is a fixed gain of the semi-blind relay R hich is given by: here C is constant. Replacing (2) in (1), then "f1"(2 "fend = C + "f2 here "fj = al / Noj, is the instantaneous SNR of the j-th hop. B. Channel Model We assume that each hop is subjected to independent, but not necessarily identically distributed, 7] -J..L fading channel, for hich the instantaneous per-hop SNR is distributed according to the PDF given by [8] as: (1) (2) (3) A. Moments of the End-to-End SNR Lemma 1: The n th -order moment of the end-to-end SNR of dual-hop semi-blind AF relaying systems is given by (6) at the top of next page. Proof See appendix A. By substituting n = 1 into (6) an expression for the average end-to-end SNR can then be obtained. Note that (6) is expressed in terms of the parameter C hich describes the relay gain. One choice for the relay gain is chosen such as [1]: c 2 =IE( 2 1 ). al + NOl By performing the required statistical average in (7), making the appropriate substitutions, and using (2), then the parameter C can be obtained as in (8) here re, ) is the incomplete gamma function [9, Eq ]. B. Outage Probability In noise limited systems, outage probability is defined as the probability that the instantaneous SNR falls belo a specified threshold "fth. Consequently, outage probability is expressed as: Pout = FYend ("(th) = Pr["fend < "fth] = Pr [ C "f "f 2 < "ftj here F'Yend ("fth) is the CDF of the "fend. Theorem 1: Let 2J..LI be natural number, the CDF of the "fend is given by (10), here Kv(-) is the modified Bessel function of the second kind of the order v [9, Eq ]. Proof See Appendix B. (7) () 2 y7rj..lj r;; J.Lj+ hj.lj j h J.Lj - 2J..Lj j (,r(j..lj)h1j - 211j+2 J, 1 1 "f exp - -_- "fj here r(-) is the gamma function [9, Eq ], 1j is the average SNR for the j-th hop, and Iv (-) is the modified Bessel function of the first kind of the order v, hich can be represented as in [9, Eq ] by: The parameter (J..Lj > 0) is the number of multi-path clusters, hj and Hj are functions of 7]j hich can be defined into to formats, namely, Format I and Format II. In this paper and ithout loss of generality, only Format I is considered. Therefore, hj = 0.25(2+ l/7]j +7]j) and Hj = 0.25(1/7]j -7]j), here (0 < 7]j < (0) represents the poer ratio of the in-phase and quadrature scattered aves in each multi-path cluster. Commonly-used fading channel models such as the Nakagami-m (J..L = m/2, 7] = 1), Rayleigh (J..L = 0.5, 7] = 1), Hoyt (or Nakahami-q) (J..L = 0.5, q 2 = 7]), and One-sided Gaussian (J..L = 0.25, 7] = 1) distributions are special cases of the generalized 7] - J..L distribution. ) (4) 4149 C. Average Bit Error Probability (ABEP) Given an expression for the MGF of the "fend, i.e., Mend(S), then using the MGF-approach described in [10] the average bit and symbol error rates for different M -ary modulations (e.g., M-PSK, M-DPSK, and M-QAM) can be evaluated. For example, the ABEP of binary DPSK (BDPSK) is given by PABEP = (1/2)Mend(1). In the sequel, e first derive an expression for the PDF of the "fend, then e use this expression to derive the MFG of the "fend Corollary 1: The PDF of the end-to-end SNR of dual-hop semi-blind AF relaying systems over generalized 7]-J..L fading channels is given by (11). Proof This PDF can be found by differentiating (10) ith respect to "fth and then by setting "fth = T The detailed proof is omitted due to space limit. Theorem 2: The MGF of the end-to-end SNR of dual-hop semi-blind AF relaying systems over generalized 7]-J..L fading channels is given by (12). Proof The MGF of the "fend can be found using: Mend(S) IE(e-S 'Yend ) = 100 f'yend(x)e-sxdx. (13) The desired result in (12) is obtained by Substituting (11) into (13), performing variable substitution y = ft, and then using [9, Eq ] and after some straightforard mathematical manipulations. Note that in (12), Wa,/3(-) represents the Whittaker function [9, Eq ].

3 (6) (8) F'Yend (rth) (10) (11) (12) 10-4 _ Rayleigh/Nakagami-m - Nakagami-m/Rayleigh 1 0-5l.::====::::=;:== = = ---.J o y, Average SNR per Hop (db) Fig. 2. Outage probability against average SNR per hop for different channel models. Ith = 0 db. IV. PERFORMANCE EVALUATION In this section, e validate our theoretical results through comparison ith Monte-Carlo simulations. We evaluate the performance of the system in terms of end-to-end average SNR, outage probability, and ABEP. The performance is studied under a ide variety of the fading parameters, average SNR per 4150 hop, and poer imbalance beteen the to hops. Any infinite series involved in the analytical computation as computed for first 25 terms (k1, k2 = 0,1,...,24). Figure 2 shos the outage probability against the average SNR per hop for different channels models. Specifically, the source-relay and relay-destination channels experience identical and mixed fading distributions, namely, Rayleigh/Rayleigh, Nakagami-m/Nakagami-m (here m = 2), Rayleigh/Nakagami-m, and Nakagami-m/Rayleigh. As mentioned earlier, these channel models are special cases of the generalized 7]-J.L fading channel and are obtained as described in Section II.B. Figure 2 is given for confirmation of correctness of our analytical derivations since it represents the results that have been already reported [1], [11]. To study the effect of the fading parameters, i.e., 7] and J.L, e plot in Figures 3 and 4 the outage probability of the system against the average SNR per hop for different values of the fading parameters. Figure 3 shos the outage performance for different values of the fading parameter 7]. It is clear from the figure that 7] has a small effect on outage performance. This is because 7] represents the poer ratio of the in-phase and quadrature scattered aves in each multi-path cluster. On the other hand, Figure 4 shos that the fading parameter J.L has a significant effect on outage probability. This is because the fading parameter J.L represents

4 -- D=0.1.M= M= D=0.5.M= = 0.8. M = l'===,---_---,---_ ---,J o y, Average SNR per Hop (db) =1i'" :=;2 =511' =251r -- :=;2 = :=;2 = O o Average SNR First Hop (db) Fig. 3. Outage probability for a dual-hop system over generalized 7] - /-i fading channels for different 7]. 'Yth = 0 db. Fig. 6. Average end-to-end SNR against 'h for 7]1 = 7]2 = 0.5, /-il = /-i2 = : Z 20 B 15. '" =0.1,)1= =0.1,)1= =0.1,)1=2 10- ;;d. _ ---=,----- o Fig. 4. Outage probability for a dual-hop system over generalized 7] - /-i fading channels for different /-i. 'Yth = 0 db. c::- l 0-1r.. =;;:;;III!Iii""' :E Rayleigh/Rayleigh, 11 = 1,11 = 0.5 g - Nakagami-m/Nakagami-m, 11 = 1, 11 = -11=1,11=1.5 ri =1,11= '1=0.5,)1=0.5 - =0.1.M=0.5 - D =0.2.M= 'n _...J.'---"-'-...:'I...J o Average SNR ofy 1 (db) alized 7] - J.L fading channels. The end-to-end SNR statistics (i.e., the CDF, PDF, and MGF) have been explicitly derived. Moreover, novel expressions for the outage probability and average end-to-end SNR are derived. The system performance is measured in terms of the end-to-end SNR, outage probability and ABEP (for BDPSK). Extending this ork to analyze the performance of AF cooperative diversity systems over generalized 7] - J.L fading channels is currently underay. ApPENDIX A PROOF OF LEMMA 1 We obtain the expression in (6) as follos: Substitute (5) into (4) then use the result in (14) at the top of next page, therefore IE(r n d) can be reritten as in (15). The first integral, II, in (15) ith respect to 11 can be solved using [9, Eq ]. The second integral in (15), hich can be reritten as in (17), can be solved as follos: using [12, Eq. lo] and [l3, Eq ], then the quantity (1 + c ;'2) -n can be expressed as: (18) and using [12, Eq. 11], the exponential function can be ritten as: (19) Fig. 5. Average BEP of a dual-hop system for different values of fading parameters. '1'2 = ')'1. the number of multi-path clusters. Therefore, as J.L increases, the number of multi-path increases, and consequently the outage performance improves. Figure 5 shos the ABEP for BDPSK modulation under different values of the fading parameters. One can clearly see that the system performance improves as the fading parameter J.L increase. In addition, the figure shos that the fading parameter 7] has a small effect on the ABEP. The reason for that is as e explained earlier. The average end-to-end SNR is plotted in Figure 6 against 1'1 by setting 7]1 = 7]2 = 0.5, J.Ll = J.L2 = 0.5, and substituting (8) into (6, ith n = 1). It can bee seen that the poer imbalance is advantageous hen 1'2 > 1'1 and it is detrimental hen 1'2 < 1'1. V. CONCLUSION In this paper, e have analyzed the performance of dual-hop semi-blind AF relay colmnunication systems, operating over independent, but not necessarily identically distributed, gener- in (10) here Gqn['l is the Meijer's G-function [9, Eq ]. Replacing (18) and (19) into (17), then (a) is obtained. Knoing that the integral of the product of a poer and to Meijer's G-function is also a Meijer's G-function [12, Eq. 21], then (b) is obtained. Note that the Meijer's G-function is available as a built-in function in many mathematical softare packages, such as MAPLE and MATHEMATICA. Finally, substituting (16) and (b) into (15), an expression for the n th -order moment of end-to-end SNR can be attained as in (6). To the best of our knoledge, (6) is ne. ApPENDIX B PROOF OF THEOREM 1 Since 11 and 12 are independent of each other, thus their joint PDF is given by (19), hich obtained using (4), (5), and after simple mathematical manipulations. Substituting (19), in the middle of next page, into (20) yields (c). The integral I3 in (c) ith respect to 11 is solved using [9, Eq ]. Using the result of I3 and binomial theorem [9, Eq ] ith condition 2J.Ll is natural number, then (21) is obtained. With the help of [9, Eq ] and [9, Eq ], e yield the desired result

5 (14) k,+2!l,+n-1 -Yl'Y1drv - 1 f(2k ) 11 e 11 - y2kl +21'1 +n 1 M1 n o 1 (15) (16) (a) 100 (1 + C ;'2) -n l k2+21l2-1 e -Y2/2d'2 (17) ft n) 100 l k2+2!l2-1 GU [ I ] G6: [Y2l2 I ] d'2 (b) C2k2+2!L2 G 1,2 [_1_ 1 1,1 + 2k2 + 2M2] f(n) 2,1 CY2 2k2 +2M2+n (19) (20) (21) REFERENCES [1] M. O. Hasna and M.-S. Alouini, "A performance study of dual-hop transmissions ith fixed gain relays," IEEE Trans. Wirel. Commun., vol. 3, no. 6, pp , [2] D. B. da Costa and M. D. Yacoub, "Dual-hop transmissions ith semi- blind relays over Nakagami-m fading channels," Electronics Letters, [10] vol. 44, no. 3, pp , [3] L. Wu, 1. Lin, K. Niu, and Z. He, "Performance of dual-hop transmissions [11] ith fixed gain relays over generalized-k fading channels," IEEE ICC, pp. 1-5, [12] [4] S. Ikki and M. H. Ahmed, "Performance analysis of dual-hop relaying communications over generalized gamma fading channels," IEEE Globe- Com, pp , [5] --, "Performance analysis of dual hop relaying over non-identical [13] Weibull fading channels," IEEE VTC, pp. 1-5, [6] A. K. Gurung, F. S. AI-Qahtani, Z. M. Hussain, and H. Alnueiri, "Performance analysis of amplify-forard relay in mixed Nakagami-m and Rician fading channels," IntI. Conf. Adv. Technol. Commun., pp ,2010. [7] M. Xia, C. Xing, y-c. Wu, and S. Aissa, "Exact performance analysis of dual-hop semi-blind af relaying over arbitrary Nakagami-m fading 4152 channels," IEEE Trans. Wirel. Commun., vol. 10, no. 10, pp , [8] M. D. Yacoub, "The 7] - J.I. distribution and the K. - J.I. distribution," IEEE Antennas Propag. Mag., vol. 49, no. 1, pp , [9] 1. S. Gradshteyn and 1. M. Ryzhik, Table of Integrals, Series, and Products. California: Academic Press, 2007, 7th ed. M. K. Simon and M. S. Alouni, Digital Communication over Fading Channels. Ne Yourk: Wiley, 2000, 2nd ed. R. Zhao and L. Yang, "Performance analysis of fixed gain relaying systems in Nakagami-m fading channels," IEEE WCSP, pp. 1-5, Y. S. Adamchik and O. 1. Marichev, "The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce systems," Proc. Int. Con! Symbolic and Algebraic Comput., pp , A. P. Prudnikov, Y A. Brychkov, and O. I. Marichev, Integrals, and Series: More Special Functions, 2007, vol. 3.