# Performance Analysis of Multihop Underground Magnetic Induction Communication
## 4 Numerical Results
### Multihop MI-WUSN Performance for INID of Hops  
To analyse the impact of the INID of the hop states in the practical deployment of an underground MI communication system, we first evaluated the variation of the end-to-end BEP with the SNR for the dual-hop scenario. For the unbalanced case, we considered that the SNR for the second hop was twice that of the first hop, that is, \(\gamma_{2}=2\gamma_{1}\). For this analysis, we used (28) to model a multihop system with the INID of hop states. Evaluation results for the dual-hop BER are shown in Figure 8. Clearly, in the case of unbalanced hops, the performance varies, unlike the balanced hop scenario. Figure 8a shows that as the noise power increases in the dual-hop system, the BER performance is degraded. This is opposite to the trend in Figure 8b, where the BER performance improves with increasing SNR. Figure 7: Multihop underground MI communication system’s performance with IID of channel states between hops. The panels show the variation of the (**a**) SNR (dB), (**b**) channel capacity (bps) and (**c**) BER with the number of hops under different noise conditions. Next, we extended this dual-hop analysis to the multihop scenario for the INID of hop states. For the analysis of the different channel states between consecutive hops as done in Section 3.3, there can be an infinite number of possibilities, which makes this analysis of NP hard complexity. To simplify this analysis and obtain meaningful results, we started from a given noise power \(P_{n}\) at the first hop and considered the following three scenarios in the analysis. 1. Increase/decrease of 0.5 dBm noise power at each hop
2. Increase/decrease of 1 dBm noise power at each hop
3. Increase/decrease of 1 dBm noise power after two hops  
The resulting graphs in Figure 9 show the average end-to-end SNR, channel capacity and BER performance across a multihop network in which the first hop is subject to an average noise power of \(-93\) dBm. The average SNR increases with the number of hops in the network in Figure 9a. This gap in system performance is not large when the deviation across hops is smaller. This behaviour can be observed in cases when there is an increase/decrease of 0.5 dBm at each hop or 1 dBm after 2 hops in Figure 9a-c. Furthermore, the rate of increase rises sharply as the state between consecutive hops changes by a greater margin; in other words, the greater the deviation across consecutive hops, the more the expected decrease in the SNR. The channel capacity in Figure 9b and the BER in Figure 9c show similar trends. It can therefore be concluded that the more the deviation of successive hop states, the more the deviation in the system performance, unlike the asymptotic model. The BER performance for the proposed INID cases also shows that a higher BER for a greater deviation can be achieved by using a greater number of hops. This observation is highlighted in Figure 9c. Once again, the need to achieve a trade-off between the number of relays and the expected system performance becomes vital for the practical deployment. The impact of the variation of the channel characteristics on the system performance is also shown in Figure 10, where hop 1 is characterized by either high (\(-83\) dBm) or low (\(-103\) dBm) noise power. The variation of this initial noise power across successive hops defines the end-to-end system performance. If the first hop is characterised by low noise, an increase of 1 dBm noise after two hops corresponds to the best end-to-end SNR performance in Figure 10a, the maximum channel capacity in Figure 10b, and the best BER performance in Figure 10c. The reverse also holds true for high noise at hop 1: after every two hops, the noise power decreases by 1 dBm. This decrease in noise corresponds  
Figure 8: Variation in the BER for balanced (\(\gamma_{2}=\gamma_{1}\)) and unbalanced (\(\gamma_{2}=2\gamma_{1}\)) dual-hop scenarios: plots of (**a**) BER vs. Noise Power (\(P_{n}(dBm)\)) and (**b**) BER vs. SNR.