## 4 Results and Discussion  
This paper presents a LLR based relay selection scheme over three state Markov channel model. MATLAB based simulation has been presented, consisting of a single antenna source, destination, and a set of relaying nodes. The simulation is carried out by considering a three-state Markovian process where the first state represents a Rayleigh fading environment, the second state represents a Rician fading environment, and the third state represents a Nakagami fading environment. To simulate system performance, four cases of transition probability have been considered. For the first case, transition probability to toggle among Rayleigh, Rician, and Nakagami fading is set to be equal; i.e., 1/3--and accordingly a transition probability matrix has been defined. The second case represents the simulation of a Rayleigh fading channel through Markov chain, transition probability moving from any state to a state representing Rayleigh fading channel is set to 1, as provided in the transition probability matrix defined for the Rayleigh fading scenario. Similarly, the transition probability matrix for a Rician and Nakagami fading scenario using a Markov chain has been created and defined through their transition probability matrix, respectively. Simulation parameters as discussed above are presented in Table 1. Furthermore, a general case of transition probability representing random variation among Rayleigh, Rician, and Nakagami fadings is also presented in Table 1 for reference purposes. Bit Error Rate (BER) and Channel Capacity are used to evaluate the system performance. Figure 2: Three-State Markov process with state-A being Rayleigh Fading, State-B represents Rician Fading and State-C represents Nakagami-m fading channel. Bit Error rate comparison for proposed LLR based relay selection scheme for a three state Markov channel is presented in Figure 3a,b. Figure 3a represents simulation for BPSK modulation scheme with no source to destination link available while Figure 3b represents a simulation for Quadrature Phase Shift Keying (QPSK) modulation. In case of BPSK modulation scheme (Figure 3a), the best BER performance is achieved when the transition probability matrix representing Nakagami fading is selected, where BER of the order of \(10^{-6}\) is achieved at 16 dB SNR value. In the case of QPSK modulation it achieves \(10^{-6}\) level at 20 dB SNR for the same parameters. Figure 4a,b presents a BER comparison for a LLR based relay selection scheme with a direct link present between the source to destination node for the BPSK and QPSK modulation scheme, respectively. For this simulation scenario, transition probabilities are set the same as previous the case and LLR is used to find the best relay node. In the case of the BPSK modulation scheme (Figure 4a) the \begin{table}
\begin{tabular}{c c} \hline
**Modulation Scheme** & **BPSK, QPSK** \\ \hline Power Allocation Factor & 0.5 \\ \hline SNR Range & 0:2:20 \\ \hline Transition Probability \(P\) (General case for Example) & [0.1 0.7 0.2; 0.0 0.4 0.6; 0.1 0.2 0.7] \\ \hline Fading environment & Three state Markov channel with Rayleigh, Rician, and Nakagami channels \\ \hline Frame Length & 256 bit \\ \hline Variance S-R, R-D and S-D link & 1 \\ \hline Transition probability matrix with equal probability & \(P=\left[\begin{array}{ccc}1/3&1/3&1/3\\ 1/3&1/3&1/3\\ 1/3&1/3&1/3\\ \end{array}\right]\) \\ \hline Transition probability matrix with Rayleigh only & \(P=\left[\begin{array}{ccc}1&0&0\\ 1&0&0\\ 1&0&0\\ \end{array}\right]\) \\ \hline Transition probability matrix with Rician only & \(P=\left[\begin{array}{ccc}0&1&0\\ 0&1&0\\ 0&1&0\\ \end{array}\right]\) \\ \hline Transition probability matrix with Nakagami only & \(P=\left[\begin{array}{ccc}0&0&1\\ 0&0&1\\ 0&0&1\\ \end{array}\right]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameters.