Performance Analysis of AF Relays with Maximal Ratio Combining in Nakagami-m Fading Environments

This paper investigates the maximal ratio combining (MRC) performance of an amplify and forward (AF) relay system in Nakagami-m fading environments.The study considers a general scenario with distinctm fading parameters for the following three links, source to relay link, and source to destination link and relay to destination link.We derive new closed form expressions for the statistics of important performance metrics, including the moment generating function, outage probability, higher order moments of equivalent signal to noise ratio (SNR), ergodic capacity, and average symbol error probability (SEP) of commonmodulation types. In particular, we focus on analytical SEP expressions in the context of an additive white generalized Gaussian noise (AWGGN). As an active area of research, generalized noise receives much attention for its flexible model. However, analytical performance of modulation scheme in generalized noise type has not been found in open literature for AF relaying with MRC despite its practical usefulness.Without the help of analytical solutions, the SEP in generalized noise can only be obtained by a large number of repeated simulation experiments. Therefore, we present the general SEP expression by using special Fox’s H function. Simulation results verify the accuracy of our theoretical analysis and show that the diversity order of MRC criterion linearly depends upon Nakagami parameters of three links.


Introduction
Maximal ratio combining (MRC) has been shown to achieve the effective reception diversity in traditional direct communications.In a relaying system, the destination can receive signals from direct source to destination link and relay link by using MRC [1][2][3].A maximum diversity order was shown in [4] for a fixed gain amplify and forward (AF) relay when MRC is used at the destination.The outage performance of zero forcing beamforming and MRC for an AF two-way underlay network was analyzed in [5] while the outage probability and approximate symbol error rate of a multiple input multiple output relay with MRC was presented in [6,7].
Although MRC has been extensively studied, most works have focused on deriving approximate solutions or upper and lower bound performance.Exact performance expressions are important in evaluating MRC but still lack sufficient study.In fact, the performance analysis of MRC encounters enormous computational complexity and inaccuracy, making it difficult to evaluate the comprehensive fading characteristics.
In this paper, we consider an AF relaying system with the MRC receiver in a Nakagami- fading environment, in which it is generally considered difficult to find the closed form expressions for performance study, especially when the links have distinct  fading parameters.In contrast to the aforementioned works, the precise performance statistics of AF system with MRC receiver are analyzed in a Nakagami- fading environment.The relay system assumes that an indirect relay link and a direct link between source and destination coexist simultaneously.We first obtain the probability density function (PDF) of equivalent signal to noise ratio (SNR) according to the convolution formula.Then the cumulative distribution function (CDF) is presented in accordance with the principle of Laplace inverse transform.New exact formulas are developed for the statistics of higher order moments of equivalent SNR, ergodic capacity, and average SEP with distinct fading parameters among different hops.In addition, our SEP expression also permits assessment of error probability in the context of an additive white generalized Gaussian noise (AWGGN) environment, which notably facilitates more general characterization of noise types.

System Model
A two-hop cooperative system with one source node S, one AF relay node R, and one destination node D is considered, as shown in Figure 1.The source node, relay node, and destination node are equipped with one antenna.Let ℎ  , ℎ  , and ℎ  represent the channel coefficients for S→R, R→D, and S→D links, respectively.Assume all the links undergo independent but not necessarily identically distributed Nakagami- fading.The instantaneous SNRs for direct S→D link and indirect relay S→R→D link are, respectively, given by where   and   denote available transmit powers of the source node and the relay node, respectively. 2  and  2  are noise variances at the relay node and the destination node, respectively.The instantaneous SNR of the MRC combiner output is given by  =  1 +  2 .The PDF of the gamma distributed variable  1 is easily given by [7]   1 () = where   is the fading severity parameter of the direct link,   denotes the corresponding scale parameters, and Γ(⋅) is the gamma function.When the fading parameter  of both S→R and R→D links is a nonnegative integer plus one half, the PDF of  2 can be found in [8] where   is the fading severity parameter,   denotes the corresponding scale parameters for  ∈ {, }, and ⌊⋅⌋ is the floor function.Since the PDF of the sum of two random variables is the convolution of the PDFs of these two variables, the PDF of MRC SNR  is thus expressed as where Although there is a simple differential and integral relationship between PDF and CDF, here one challenge arises from the generalized hypergeometric function when integrating the above PDF.Thus we turn to moment generating function.Using [9, eq.(3.35.3)], the moment generating function of  is given by Further, by using [10, eq.(2.1.3.1)], the CDF of  is given by where  −1 (⋅) is the Laplace inverse transform and Φ 2 (⋅, ⋅; ⋅; ⋅, ⋅) is the confluent hypergeometric function of two variables [11].

Outage Probability and SNR Moments.
Outage probability is the probability that the SNR fails to meet a predetermined threshold and mathematically expressed as where  ℎ is the threshold value.
According to the relationship between the moment generating function and the higher order moments, the th order moment of SNR  is given by where E[⋅] is expectation operation,  ()  () is the th differential with respect to   (), and ()  is the Pochhammer symbol [12].

Average SEP.
For some simple modulation constellations, there is only one parameter carrying information such as phase, frequency, and amplitude.The error probability of these modulations is usually related to only one Gaussian  function, such as binary phase shift keying and pulse amplitude modulation.However, for some complex modulations, there are two or more parameters carrying signal information, so the error probability of data transmission involves the square of the Gaussian  function, such as quadrature amplitude modulation (QAM).The average SEP is expressed in a unified form as follows: where , , and  are constants, whose values are dependent on specific modulation constellation.For a simple modulation, only a single Gaussian  function is included in error probability and  = 0 is set in (15).Similar to the case of ergodic capacity, it is a little troublesome to calculate the error probability directly.Because of the complexity of the Gaussian  function, we start with the first part in (15).
Capitalizing [16, eq.( 33)], the average SEP can be rewritten in an alternative form as where (⋅) is Meijer's G function.Utilizing standard integration by parts, after some mathematical rearrangements, we will encounter a type of integral as follows: Substituting   () into ( 17) yields an integral given by Wireless Communications and Mobile Computing 5 Making a change of a variable  = 1/ leads to Similarly, in ergodic capacity case, by Fox's  functions, the power function is represented as Capitalizing the integral table [15, eq.(2.5.1)],  4 can be expressed in bivariable Fox's function.Finally, substituting the integral result into (15) and solving the remaining algebraic operation yields average SEP containing only one single Gaussian  function given by Wireless Communications and Mobile Computing Next, we focus our attention on the second part in (15) involving the square of Gaussian  function.According to moment generating function approach, we have Utilizing [17, eq.( 21)] and after some mathematical manipulations,  2 is written in where  ()  is Lauricella's function of the fourth kind [18, eq.( 15)].Finally, the average SEP is fully expressed as   =  1 −  2 .
In ( 21) and ( 23), we derive the SEP for the AF relay system, but it is possible to find the SEP of a little bit more generalized case.Recently, some researchers pointed out that white Gaussian noise is not a suitable noise model in some special scenarios, such as wireless sensor networks and underwater communication [19].The generalized Gaussian white noise model is the best choice.However, few works have considered SEP in the context of generalized noise environment in a relay system.Under the AWGGN environment, the average SEP is given by where   (⋅) is the generalized  function defined as and Λ 0 = √Γ(3/)/Γ(1/).Different  parameters correspond to different noise environments.The corresponding relationship between  parameter and noise environment is summarized in [19].The generalized SEP analysis is a complicated problem due to the involvement of generalized  function.To the best of our knowledge, this is the very first effort for this considered setup.The major challenge for this extension arises from the generalized  function.Substituting   () into the average SEP yields a type of integral: By Fox's  function, the incomplete gamma function and hypergeometric function are, respectively, expressed as ] Utilizing [15, eq.(2.6.2)], 5 can be solved in a closed form by bivariable Fox's function.Thus the average SEP is given by Note ( 29) is reasonable only when   < (√  + √  ) 2 due to the convergence property of bivariable Fox's function.
Hitherto, all the performance expressions in this paper are established in the closed forms.These formulas significantly reduce the computational complexity and also greatly facilitate the general performance study in a convenient manner.As mentioned earlier, in the existing literature, the performance expressions either remain in an integral form with an integrand of moment generating function or only provide a lower bound and/or an upper bound.

Simulation Results
Numerical performance results are provided through both closed form analytical expressions and simulations.For simplicity,   =   is assumed and   / 2  and   / 2  are defined as average SNR per hop.The channel gain is normalized to unity unless otherwise noted.
Figure 2 shows the tightness of the analytical curves on the MRC outage performance compared with simulation results over the whole range of SNR, where weak SD means poor quality of S→D link; i.e., E[|ℎ  | 2 ]/E[|ℎ  | 2 ] = -10dB.Due to the poor quality of direct channel, the outage probability of weak SD is higher than that of good quality.The diversity order can be obtained through a wide range of channel settings.For example, in case of   = 0.5,   = 1.5, and   = 0.1234 with weak SD,   = 0.0301 at 28dB and   = 0.0226 at 30dB.Thus, the diversity impact on a high SNR slope of the outage curve is quantified by 10log 10 (0.0301/0.0226)/(30 − 28) = 0.6223 ≈ min(  ,   ) +   .Simulation results demonstrate that MRC attains a diversity order of   + min(  ,   ).
Figure 3 compares the first-order moment of the SNR with various fading parameter settings.We can see that the analytical average SNR accurately matches simulation result in all fading parameters.A constant gap exists between the weak SD channel and the good channel due to the poor quality in weak SD channel.
Figure 4 shows the ergodic capacity.As expected, the theoretical curves precisely match with simulation results in all SNRs.In the high SNR regime, the capacity curves demonstrate clearly the multiplexing gain performance.For example, when   =   = 0.5 and   = 1.5678, the ergodic capacities are 4.6239 at 28dB and 4.9554 at 30dB, respectively.So the multiplexing gain is quantified by 10 log 10 2 × (4.9554 − 4.6239)/(30 − 28) = 0.4990 ≈ 1/2.
Figure 5 depicts the average SEP of 4QAM as a function of SNR.And, as a comparison, the SEP of the higher order constellation is shown in Figure 6, where   = 0.5,   = 1.5, and   = 1.5678.From both figures, an accurate diversity order agreement between analysis and simulation is found in the high SNR regime.This can be explained by the fact that outage probability and average SEP both achieve the same diversity order   + min(  ,   ).In Figure 6, all SEP curves follow the same downward trend and become approximately parallel lines in the high SNR region, indicating the same diversity order, independent of the specific constellation.The first order moment of the SNR

Conclusion
This paper studies the performance of AF relaying with MRC in Nakagami- channels.New closed form formulas are derived for CDF, moment generating function, outage probability, higher order moments of the SNR, ergodic capacity, and average SEP and greatly facilitate the general performance study in a convenient manner in such a system.The theoretical expressions coincide with the simulation results precisely under various system settings.