# On Multi-Hop Decode-and-Forward Cooperative Relaying for Industrial Wireless Sensor Networks
## 5 Numerical Results
### Results on Channel Capacity  
Figures 9 and 10 sketch the end-to-end average capacity of the relay network employing the full-duplex communication mode and balanced power allocation scheme as a function of the average SNR for varying numbers of hops \(L\). It can be seen that excellent fit is found between the analytical results from the proposed closed-form expression and the simulation results. Figure 9 shows the channel capacity of relay network under the Nakagami-\(m\) fading channels (\(m=3\)) in the presence of mediumly impulsive noise with \(A=0.5\) and \(\rho=1\). Figure 10 shows the channel capacity of the relay network under Rayleigh fading channels (\(m=1\)) in the presence of nearly Gaussian noise with \(A=10\). As expected, the overall channel capacity degrades as \(L\) increases in exchange for broader transmission coverage. A less impulsive channel (i.e., when \(A\) or \(\rho\) are large enough such that the MCA channel degenerates to the AWGN channel) gives a capacity approaching that of the AWGN channel. Since AWGN is known to be the worst additive interference in terms of channel capacity for both point-to-point channels and relay channels, all values of the different MCA model parameter sets provide higher capacity than the corresponding AWGN channel [48]. However, it should be noted that the AWGN channel does not necessarily underperform the corresponding MCA channel in terms of BER, as can be seen from Figure 7.  
The PDFs of the end-to-end instantaneous capacity of the multi-hop relay system under Rayleigh (\(m=1\)) and Nakagami-\(m\) (\(m=3\)) fading channels are presented in Figure 11. The following parameters are considered: the impulsive index \(A=0.5\), the ratio \(\rho=1\), and the average SNR being 10 dB. It can be found from Figure 11 that an increase of the severity of large-scale fading (i.e., decreasing the value of the Nakagami-\(m\) parameter \(m\)) decreases the mean value of the channel capacity. Similarly, an increase of the number of hops \(L\) also degrades the mean channel capacity. These can also be evidenced by comparing Figures 9 and 10. It can also be observed that a decrease in the parameter \(m\) or the number of hops \(L\) results in an increase of the variance of the channel capacity, which will be reconfirmed in Figure 12.  
Figure 9: The channel capacity of Nakagami-\(m\) channels (\(m=3\)) with \(L=1,2,5,8\) relay hops.  
Figure 12 illustrates the behavior of the variance of the channel capacity with varying average SNR per link. We calculate the variance for different number of hops and varying values of the Nakagami \(m\) parameter. It is shown that the variance increases as the average SNR increases. More specifically, it is found that the variance increase monotonically from low SNR to high SNR region and afterward it continues to maintain approximately the same level in the high SNR region. The variation of the channel capacity is also found to increase with the reduction of the Nakagami \(m\) parameter or increase of the number of relay hops.  
Figure 11: The PDF of channel capacity of Rayleigh channels (\(m=1\)) and Nakagami-\(m\) channels (\(m=3\)) with \(L=8,5,2\) relay hops.  
Figure 10: The channel capacity of Rayleigh channels (\(m=1\)) and nearly AWGN (\(A=10\)) with \(L=1,2,5,8\) relay hops.