Abstract
We present a new analytical expression for the moment generating function (MGF) of the endtoend signaltonoise ratio of dualhop decodeandforward (DF) relaying systems with relay selection when operating over Nakagami 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 ary phase shift keying modulation for the considered DF relaying scheme under various asymmetric fading conditions. It is shown that the MGFbased 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.
1. 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 [1–3]. One of the bandwidth efficient dualhop cooperative techniques combines the decodeandforward (DF) relaying protocol with relay selection (RS) [4]. The performance of this technique has been studied in [4] over Rayleigh fading channels and in [5–7] for the more general Nakagami fading channel model. However, in the latter works analytical expressions for the moment generating function (MGF) of the endtoend signaltonoise 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 endtoend SNR presented in [8], whereas [6, 7] utilized the endtoend 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 closedform representation for the MGF of the endtoend SNR of dualhop DF RSbased systems which is valid for arbitraryvalued 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 closedform representation for the MGF of DF with RS over arbitrary Nakagami fading channels is derived in Section 3. An application of the derived expression in evaluating the ASEP performance of PSK is demonstrated in Section 4, while closing remarks are provided in Section 5.
2. Signal and System Model
Consider a wireless dualhop DF cooperative system comprising a source node , a destination node , and halfduplex relays, each of which is denoted by , . During the first transmission time slot, broadcasts a symbol to the relays. In the second time slot, the relay with the most favorable channel to is selected to transmit, while remains silent. Node is assumed to possess perfect channel state information so that maximumlikelihood combining of the signals from and the selected relay can be employed.
The wireless channels between any pair of system nodes are assumed to be subject to the Nakagami fading distribution [13]. Furthermore, transmission is corrupted by additive white Gaussian noise (AWGN) with singlesided power spectral density . The fading parameters for the , , and channels are denoted by , , and , respectively, whereas the corresponding instantaneous received SNRs as , , and , respectively. It is noted that unlike many other studies [4–7], no restrictions are applied on the Nakagami fading parameters of both hops. In other words, and each , with and , can take any arbitrary positive real value as long as it obeys the necessary condition for the Nakagami distribution, that is, . The average received SNRs at the first and second hop are defined as , , and , respectively, where denotes the average fading power of the channel and is the energy of the symbol transmitted. Clearly, and ’s are Gamma distributed random variables which are assumed to be statistically independent.
As shown in [8], a dualhop transmission employing the DF protocol can be modeled as a singlehop transmission with the equivalent instantaneous received SNR at bounded as In (1), and denotes the transmitted signal’s modulation order. For relatively large SNR and values, setting yields a tight approximation for [8]. To this effect, by using RS, the corresponding received SNR at can be accurately approximated as [5]. Utilizing next a timediversity version of maximalratio combining [11], the instantaneous received SNR at ’s output can be expressed as
3. MGF of DF RS in Nakagami Fading
By differentiating [5, equation ], a closedform expression for the probability density function (PDF) of for any arbitrary value of can be obtained as follows: where , , and , with , while and denote the incomplete [14, equation ] and the upper incomplete Gamma function [14, equation ], respectively. Substituting [5, equation ] and (3) into [5, equation ] and after some algebraic manipulations, the following expression for the PDF of is deduced: where symbol is used for shorthand representation of multiple summations of the form with .
In order to derive an explicit expression for the MGF of , which is defined as [11, equation ], for arbitrary valid values of both and , one is required to analytically evaluate integrals that involve combinations of arbitrary powers, exponentials, and functions. To this end, by expressing all ’s in (4) according to [14, equation ], utilizing [14, equation ], and performing some rather long but basic algebraic manipulations, one obtains the following explicit expression for : which is valid for for all . In (5), parameters and , with , are given by In (6)(7) the notation is used for shorthand representation of and , and symbols and represent and , respectively, whereas the function of denoted by , with and , , and , is defined as
Importantly, after careful inspection of ’s and ’s for in (5), it follows that they can all be expressed in terms of wellknown generalized hypergeometric functions [14, equation ]. To this end, for the ’s and ’s with , , and , by expressing every term of the form as and using the identity , where is the Pochhammer symbol [14, page xliii], yields where is given by In (10), represents the Gauss hypergeometric function [14, equation ]. Likewise, for the ’s and ’s with , , and , each infinite series can be ultimately expressed in terms of the Appell hypergeometric function [14, equation ] as where , with , is given by Finally, by using once more all aforementioned identities, it follows that the and coefficients can be expressed in terms of the generalized Lauricella function [15, equation (1.1)], yielding To this effect, substituting (6), (9), (11), and (13) to (5), a novel expression for is deduced which is valid for arbitrary . It is noted that in Appendices A and B we present MATLAB routines for computational efficient implementations of functions and , respectively.
Using the previously derived expression, the MGF of and by recalling that and are statistically independent, the MGF of is straightforwardly deduced, namely,
4. ASEP Performance Evaluation
The ASEP performance of various modulation schemes for the considered dualhop DF relaying system with RS over Nakagami fading channels can be directly evaluated using the expression and the MGFbased approach presented in [11, Chapter 1]. For example, the ASEP of PSK modulation is easily obtained as
Let us assume a relaying system with relays, no direct link , and the following four asymmetric fading scenarios: (i) Scenario A: , , , and for , , and ; (ii) Scenario B: , , , and for , , and ; (iii) Scenario C: , , , and for , , and ; and (iv) Scenario D: , , , and for , and . The ASEP performance, , of ,  and PSK modulation is depicted in Figures 1, 2, and 3, respectively, as a function of the average symbol to noise power, , over various Nakagami fading conditions. As expected, improves with increasing 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 , the more sensitive is to slight variations of the fading conditions. More specifically, for the case of PSK modulation, it can be observed that the differences in the curves among the four considered fading scenarios for the low regime, for example, at dB, are between Scenarios A and B; between Scenarios B and C; and between Scenarios C and D. For the same modulation order, the differences in the high regime, for example, at dB, are: between Scenario A and B; between Scenario B and C; and between Scenario C and D. Likewise, for the case of PSK and for dB, the differences in the curves for the different scenarios are between Scenarios A and B; between Scenarios B and C; and between Scenarios C and D. For PSK modulation and for dB, the differences among the various curves are between Scenarios A and B; between Scenarios B and C; and between Scenarios C and D. Finally, for PSK modulation, it is evident that the curves differ at dB: between Scenarios A and B; between Scenarios B and C; between Scenarios C and D whereas, for dB, the differences among scenarios are: between Scenarios A and B; between Scenarios B and C; and 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 considered communication scenarios and particularly in the case of modulation schemes with low and/or high values.
5. Conclusion
This work was devoted to the derivation of a new analytical expression for the MGF of the endtoend SNR of dualhop 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 MGFbased 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.
Appendices
A. MATLAB Program for the Evaluation of
The Appell hypergeometric function is defined by the double infinite series given by [14, equation ]. An equivalent expression for in a double integral form with finite limits is given by [14, equation ]. The latter expression can be easily evaluated using the MATLAB function shown in Algorithm 1.

B. MATLAB Program for the Evaluation of
The generalized Lauricella function is expressed according to the triple infinite series given by [15, equation ]. With the aid of the integral representation for in [14, equation ], the following triple integral form for with finite limits is deduced: where and To this effect, the MATLAB function shown in Algorithm 2 evaluates using (B.1).

Acknowledgments
The work of Dr. George C. Alexandropoulos has been partly supported by the General Secretarial Research and Technology (G.S.R.T.), Hellas, under the ARTEMIS research program R3COP with the Project number 100233. The work of Dr. Khuong HoVan is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 102.042012.39.