Performance of wireless communication suffers from different fading effects and these effects can be reduced by using multiple-input multiple-output (MIMO), or relayed transmission technique. Consequently, this leads to a great research interest in the MIMO and the relayed communications techniques. In amplify-and-forward (AF) relayed transmission technique, a wireless link has been established between the source and the destination via the relay. Here the source is transmitting a signal to the relay (known as 1^st hop S-R) and relay detects the signal with amplification sending it to the destination (known as 2^nd hop R-D) [1] . On the accessibility of instantaneous channel sate information (CSI) at the relay node, AF relaying scheme is generally categorized into two parts, one is variable-gain relaying [2] and the other is fixed gain AF relaying [3] . In such communication networks, the signal may be affected by different types of fading. Taking the effect of fading in account, we can divide AF relay network into two categories, one is for symmetric fading channels and the other is for asymmetric fading channels. In the past years, the performance of relay networks (two ways, dual hope, multi hope, and multi cast) over symmetric fading channels is being carried out by many researchers [4] . The performances of dual hop wireless link under symmetric (Nakagami-m) fading channels have been discussed in [5] . However, in practical scenario, relay network may have line of sight (LOS) communication from source to destination and multipath communication from relay to destination and vice versa.

In present days the research interest on relay network over asymmetric fading channels has been increased and some recent examples of that are [6] -[8] . For dual hop AF relay transmission system, the performance of the network has been studied in [7] by deriving analytical expression for the cumulative distribution function (cdf) and the probability density function (pdf) of end to end signal to noise ratio (SNR) over asymmetric (Nakagami-m and Rician) fading channels. Here authors have studied the performance of beam forming in dual hop AF network and have derived closed-form cdf, pdf and expressions of the received SNR and average symbol error rate (SER) for Rayleigh fading channel. The outage probability and the average bit error probability have been studied in [8] for asymmetric (Rayleigh and Rician) fading channels. In both cases ergodic capacity analysis has been taken into consideration. In [9] , closed-form cdf, pdf and moment generating function (MGF), expressions of received SNR and SER have been derived for MIMO AF relay network over a Rayleigh fading channel, where transmit antenna selection and maximal ratio combining are used at the source and the destination, respectively, and the relay is equipped with a single antenna [10] . In such an asymmetric fading scenario, the above studies are promising for improving the performance of a wireless communication link.

In this paper, we assume dual hope AF relay network over symmetric and asymmetric fading channels and study the performance of the relay network by evaluating the closed form expressions for outage probability, average SER by using cdf and ergodic capacity by using pdf. We plot the analytical results for the variation of combined cdf and pdf against the instantaneous SNR considering different symmetric (Nakagami-Nakagami, Releigh-Rayleigh) and asymmetric (Nakagami-m-Rayleigh, Rayleigh-Nakagami-m) fading channels for a relay network. Finally, the effects of these fading channels on the relay network are shown by considering the variation of SER against SNR of the 2^nd hop. It is to be mentioned here that in two-hop wireless link, there is a good possibility of experiencing different fading environments in first and second hop. Most of the papers claim their contribution in deriving the expression of SER and Outage probability in closed forms from different angles, where the expressions become very complicated. Here, we are only interested to find the solution in a simpler method.

The paper is organized as follows: Section 2 provides the theoretical analysis in determination of the combined pdf and cdf of 2-hop wireless link along with SER, Section 3 deals with the results based on analysis of Section 2 and finally Section 4 concludes the entire analysis.

In a 2-hop wireless link sometimes the 1^st hop (S-R) and the 2^nd hop (R-D) experience different types of fading called asymmetric or mixed fading environment as discussed in [4] [5] [7] [8] and [11] . Let the pdf of the SNR

of 1^st hop is and that of 2^nd hop is; where and are the average SNRs of the

corresponding links. The cdf of the combined link is expressed [11] :

where

and

.

Equation (1) can then be written as

where is the cdf of 1^st link.

If S-R link is under Nakagami-m fading then its pdf

Based on [4] [7] [8] the cdf will be,

From Equations (2) and (4),

If the R-D link is under Rayleigh fading then,

From Equations (5) and (6), we have

From [5] , for the condition, the range of will be. Again for the condition, , the range of will be. Hence, Equations (4) to (7) has to be changed accordingly.

Putting is equal to threshold SNR in Equation (7) we get the outage probability. Differentiating Equation (7), we get the pdf of the combined link as

The BER can be determined according to [12] as

where the magnitude of depends on the particular modulation scheme (for example for QPSK, for BPSK, etc.).

The 2^nd formula for SER is

The 3^rd formula for SER [13] is

It is to be mentioned here that, for M-ary-QAM and.

The variation of combined cdf and pdf against the instantaneous SNR are shown in Figures 1-4 (four combinations of fading) taking average SNR (absolute) of the 1^st hop and the 2^nd hop as a parameter. Both cdf and pdf are found stepper for lower values of SNR-1 and SNR-2 for all the four combinations of fading. The profile of pdf and cdf are all exponential but the pdf of Nakagami-Nakagami and Nakagami-Rayleigh combinations at higher average SNR provides pdf like Poisson’s pdf is shown in Figure 2 and Figure 3. With the help of both pdf and cdf, we determine the SER of 2-hop link with mixed fading under QPSK. The variation of SER against average SNR of 2^nd hop is shown in Figure 5 taking average SNR of 1^st hop of 5 dB, m₁ = 2, and m₂ = 4. The performance is found the best for Nakagami-Nakagami combination and worst for Rayleigh-Rayleigh combination. In Nakagami-m fading, there are m direct link between transmitter and receiver hence the results of Figure 5 is obvious. We also include the SER of single hop link under both Rayleigh and Nakagami-m fading case in Figure 5 to get the idea about the amount degradation of 2-hop link for 4 possible fading combinations. We have to allude one thing that if the length of the single hop link is equal or greater that the 2-hop link then the performance of single hop will be much worse than the case of 2-hop link. Our next step is to make the 1^st hop stronger taking SNR of 1^st hop of 12 dB, m₁ = 2, and m₂ = 4. The corresponding profile of SER is shown in Figure 6. This time the performance of the network is found much better than the case of Figure 5. In this case stronger signal will be received by the relay hence the received signal will maintain SNR greater than the threshold. Therefore, the impact of pdf of 1^st hop is more prominent than that of 2^nd hop.

Finally, the performance of mixed 2-hop (1^st hop is Nakagami-m, m = 4 and 2^nd hop is Rayleigh) wireless link is shown in Figure 7 in context of outage probability vs. average SNR of 2^nd hop. The performance of the communication system enhances with increments of average SNR of 2^nd hop and with the decrement of the threshold SNR of 1^st hop is visualized from Figure 7.

In this paper we deal with the performance of 2-hop wireless link under asymmetric fading environment. Here, we have implemented the analytical model of asymmetric fading in a simpler way. We have combined the pdf of fading and the environment of AWGN to get the SER and Outage probability under asymmetric fading condition. Major applications of such network lie in: wireless ad-hoc network, wireless sensor network and virtual MIMO of 5G network. Here entire analysis is done under single antenna case. The work can be enhanced with multiple antennas on transmitting or receiving end; even on both sides. In this case we can apply the technique of TAS (transmitting antenna selection) on transmitting side or combining scheme on receiving side. Still we can apply zero forcing equalization technique or match filter on receiving end to improve the received SNR.

SarwarJahan,Md ImdadulIslam,Mohamed RuhulAmin, (2015) Performance Evaluation of Two-Hop Wireless Network under Asymmetric Fading Environment. Journal of Computer and Communications,03,21-27. doi: 10.4236/jcc.2015.312003