Symbol Error Probability of DF Relay Selection over Arbitrary Nakagami-m Fading Channels

We present a new analytical expression for the moment generating function (MGF) of the end-to-end signal-to-noise ratio of dual-hop decode-and-forward (DF) relaying systems with relay selection when operating over Nakagami-m fading channels. The derived MGF expression, which is valid for arbitrary values of the fading parameters of both hops, is subsequently utilized to evaluate the average symbol error probability (ASEP) of M-ary phase shift keying modulation for the considered DF relaying scheme under various asymmetric fading conditions. It is shown that the MGF-based ASEP performance evaluation results are in excellent agreement with equivalent ones obtained by means of computer simulations, thus validating the correctness of the presented MGF expression.


Introduction
Cooperative communication through relay nodes has been shown to be capable of extending the radio coverage and improving the reliability of emerging wireless systems [1][2][3].One of the bandwidth efficient dual-hop cooperative techniques combines the decode-and-forward (DF) relaying protocol with relay selection (RS) [4].The performance of this technique has been studied in [4] over Rayleigh fading channels and in [5][6][7] for the more general Nakagami- fading channel model.However, in the latter works analytical expressions for the moment generating function (MGF) of the end-to-end signal-to-noise ratio (SNR) have been presented, which are valid only for the special case where the Nakagami -parameter of both hops takes integer values.In [5], the authors based their analysis on the tight approximation for the end-to-end SNR presented in [8], whereas [6,7] utilized the end-to-end SNR characterization of [9].Nevertheless, in realistic wireless communication scenarios, estimators for  from field measurement data typically result in arbitrary  values [10].Moreover, restricting  to only integer values severely limits the advantageous property of the Nakagami- fading distribution to adequately approximate the Rice and Hoyt ones [11].Very recently, based on [9], the authors in [12] investigated the error probability of opportunistic DF relaying over Nakagami- fading channels with arbitrary .
In this paper, capitalizing on the approach of [8], we present a new closed-form representation for the MGF of the end-to-end SNR of dual-hop DF RS-based systems which is valid for arbitrary-valued parameters for both Nakagami- faded hops.In addition, the derived expression is utilized to evaluate the average symbol error probability (ASEP) of ary phase shift keying (PSK) modulation for the considered relaying scheme.Numerically evaluated ASEP results match perfectly with equivalent results obtained from computer simulations and clearly demonstrate that ASEP is rather sensitive to even slight variations of any of the hops' fading parameter.
The remainder of this paper is organized as follows.Section 2 presents in brief the corresponding signal and system model.A new closed-form representation for the In (1),  min = min{ 1 ,  2 } and  denotes the transmitted signal's modulation order.For relatively large SNR and  values, setting   =  min yields a tight approximation for   [8].To this effect, by using RS, the corresponding received SNR at  can be accurately approximated as  RS = max =1,2,...,   [5].Utilizing next a time-diversity version of maximal-ratio combining [11], the instantaneous received SNR at 's output can be expressed as

MGF of DF RS in Nakagami-𝑚 Fading
By differentiating [5, equation (3)], a closed-form expression for the probability density function (PDF) of   for any arbitrary value of  1 ,  2 ≥ 0.5 can be obtained as follows: where ] and (3) into [5, equation (6)] and after some algebraic manipulations, the following expression for the PDF of  RS is deduced: where symbol ∑ is used for short-hand representation of multiple summations of the form In order to derive an explicit expression for the MGF of  R , which is defined as for arbitrary valid values of both  1 and  2 , one is required to analytically evaluate integrals that involve combinations of arbitrary powers, exponentials, and (⋅, ⋅) functions.To this end, by expressing all G(⋅, ⋅)'s in (4) according to [14, equation (8.354.2)],utilizing [14, equation (8.310)], and performing some rather long but basic algebraic manipulations, one obtains the following explicit expression for   RS (): which is valid for  1 ,  2 ̸ =  for all  = 1, 2, . . ., .In (5), parameters A  and B  , with  = 1, 2, . . ., 8, are given by In ( 6)-( 7) the notation {   } = {   } is used for short-hand representation of  =  and  = , and symbols Ξ and Υ represent {  2  1 } and {  1  2 }, respectively, whereas the function of  denoted by   (, , ), with , ,  ∈ R and  = 0, 1, 2 and 3, is defined as Importantly, after careful inspection of A  's and B  's for  = 2, 3, . . ., 8 in (5), it follows that they can all be expressed in terms of well-known generalized hypergeometric functions [14, equation (9.14)].To this end, for the A  's and B  's with  = 2, 3, and 5, by expressing every term of the form ( + ) as ( + )!/( +  − 1)! and using the identity Γ( + ) = ()  Γ(), where ()  is the Pochhammer symbol [14, page xliii], yields where (, ) is given by In (10) where (, , , ]), with ] ∈ R, is given by Finally, by using once more all aforementioned identities, it follows that the A 8 and B 8 coefficients can be expressed in terms of the generalized Lauricella function  (3)   (⋅) [15, equation (1.1)], yielding To this effect, substituting (6), ( 9), (11), and (13) to (5), a novel expression for   RS () is deduced which is valid for arbitrary  1 ,  2 ≥ 0.5.It is noted that in Appendices A and B we present MATLAB routines for computational efficient implementations of functions  2 (⋅) and  3  (⋅), respectively.Using the previously derived   RS () expression, the MGF of  0 and by recalling that  RS and  0 are statistically independent, the MGF of   is straightforwardly deduced, namely,

ASEP Performance Evaluation
The ASEP performance of various modulation schemes for the considered dual-hop DF relaying system with RS over Nakagami- fading channels can be directly evaluated using the    () expression and the MGF-based approach presented in [11,Chapter 1].For example, the ASEP of -PSK modulation is easily obtained as Let us assume a relaying system with  = 3 relays, no direct link  → , and the following four asymmetric fading scenarios: (i) Scenario A:  1, 2, and 3, respectively, as a function of the average symbol to noise power,   / 0 , over various Nakagami- fading conditions.As expected,  se improves with increasing   / 0 and/or decreasing  and/or increasing any of the fading parameters.Furthermore, all figures clearly depict the excellent agreement between the  numerically evaluated ASEP results and the equivalent ones obtained from Monte Carlo simulations.In addition, it is observed that the higher is the   / 0 , the more sensitive is  se to slight variations of the fading conditions.More specifically, for the case of 8-PSK modulation, it can be observed that the differences in the  se curves among the four considered fading scenarios for the low   / 0 regime, for example, at 5 dB, are 51% between Scenarios A and B; 52% between Scenarios B and C; and 54% between Scenarios C and D. For the same modulation order, the differences in the high  s / 0 regime, for example, at 20 dB, are: 98% between Scenario A and B; 100% between Scenario B and C; and 103% between Scenario C and D. Likewise, for the case of 16-PSK and for   / 0 = 5dB, the differences in the  se curves for the different scenarios are 23% between Scenarios A and B; 29% between Scenarios B and C; and 36% between Scenarios C and D. For 16-PSK modulation and for   / 0 = 20 dB, the differences among the various  se curves are 84% between Scenarios A and B; 90% between Scenarios B and C; and 98% between Scenarios C and D. Finally, for 32-PSK modulation, it is evident that the  se curves differ at   / 0 = 5dB: 11% between Scenarios A and B; 14% between Scenarios B and C; 19% between Scenarios C and D whereas, for   / 0 = 20 dB, the  se differences among scenarios are: 46% between Scenarios A and B; 62% between Scenarios B and C; and 79% between Scenario C and D. It is evident from the above quantitative results that the sensitivity of the ASEP on the fading severity parameter  is high in all Function F2 = Appell(a,b1,b2,c1,c2,x,y); f1 = gamma(c1).* gamma(c2); f2 = gamma(b1).* gamma(b2); f3 = gamma(c1-b1).* gamma(c2-b2); f = f1./(f2.* f3); Q = @(u,v)f.* u.̂(b1-1).* (v.̂(b2-1)).* ...

Conclusion
This work was devoted to the derivation of a new analytical expression for the MGF of the end-to-end SNR of dual-hop DF relaying communication systems with RS over Nakagami- fading conditions.The presented expression involves wellknown generalized hypergeometric functions and is valid for any arbitrary value of the fading parameters of both hops.Using the MGF-based approach, the ASEP of -PSK modulation for the considered system was evaluated and a perfect match with equivalent computer simulated performance results was shown.More importantly, it was evident that the ASEP is sensitive to even slight variations of any of the hops' fading parameters and particularly for low order modulation schemes and high SNR values.

Figure 1 :
Figure 1: Average symbol error probability,  se , of 8-PSK of dualhop DF relaying with RS as a function of the average symbol to noise power,   / 0 , for  = 3 relays and various Nakagami- fading conditions.

Figure 2 :
Figure 2: Average symbol error probability,  se , of 16-PSK of dualhop DF relaying with RS as a function of the average symbol to noise power,   / 0 , for  = 3 relays and various Nakagami- fading conditions.

Figure 3 :
Figure 3: Average symbol error probability,  se , of 32-PSK of dualhop DF relaying with RS as a function of the average symbol to noise power,   / 0 , for  = 3 relays and various Nakagami- fading conditions.