# Performance Analysis of Multihop Underground Magnetic Induction Communication
## 4 Numerical Results
### Single-Hop MI-WUSN Performance  
Figure 5 shows the result of a performance analysis of point-to-point communication performed with the single-hop model of Section 3.1. The results in Figure 5a,b can be recreated using (8) and (11), respectively. Figure 5c is based on (12) and (13). Figure 5a shows that the SNR for high noise power (\(-83\) dBm) is lower than that for low noise power (\(-103\) dBm) and decreases with increasing transmission distance. The channel capacity in Figure 5b is lower for high noise power and decreases with increasing transmission distance. The BER performance in Figure 5c shows that binary phase-shift keying (BPSK) is a better modulation scheme than 2 ASK for both low and high noise scenarios. Accordingly, further analysis in this study was performed for BPSK modulation.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Operational frequency, \(f\) & 10 MHz \\ Transmission distance, \(\mathcal{R}\) & 25 m \\ Number of nodes, \(k\) (with K–2 relays) & 0–12 \\ Number of turns, \(N\) & 5 \\ Coil radius, \(a\) & 0.15 m \\ Resistance per unit length, \(R_{0}\) & 0.01 \\ System Bandwidth, \(W\) & 2000 Hz \\ Noise power, \(P_{n}\) & \(P_{n(min)}=-103\) dBm, \(P_{n(max)}=-83\) dBm \\ Magnetic Constant, \(\mu_{0}\) & \(4\pi\times 10^{-7}\) H/m \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters used for evaluating the wireless underground MI sensor network system’s performance.  
Figure 4: Performance analysis methodology of multihop underground magnetic induction communication system in terms of SNR, channel capacity and BER. For a given transmission distance, performance evaluation has been carried out for three types of networks: (1) single hop, (2) multihop with independent and identical distribution (IID) of hop states, and (3) multihop with independent and non-identical distribution (INID) of hop states.  
To emphasise the significance of an MI-based underground communication system beyond asymptotic analysis based on static channel conditions, Figure 6 shows the variation of the achievable BER with noise in the link for a given SNR. It can be clearly seen that for the achievement of a reference BER of 0.25, internode distances of around 3, 5 and 8 m are required for high noise power (\(-83\) dBm), average noise power (\(-93\) dBm) and low noise power (\(-103\) dBm), respectively. Alternatively, the optimal internode distance can increase from 3 m to 8 m as the noise decreases from \(-83\) dBm to \(-103\) dBm. We can therefore deduce that the results of an earlier work based on an asymptotic analysis for MI-WUSN design in which the channel properties were considered to be constant are not applicable to multihop analysis, since a variation in the any of the links can change the system performance. This implies that system design for best/worst case scenarios is not practically feasible so smart design optimizations need to be analysed.  
Figure 5: Single-hop underground MI communication performance. The panels show the variation in the (**a**) signal-to-noise ratio (SNR (dB)), (**b**) channel capacity (bps) and (**c**) bit error rate (BER) over the transmission distance at low noise power (\(P_{n}=-103\) dBm) and high noise power (\(P_{n}=-83\ dBm\)) for the binary phase shift keying and binary amplitude shift keying modulation schemes.