# Performance Analysis of Multihop Underground Magnetic Induction Communication
## 3 Performance Analysis
### Single-Hop MI Communication Model  
From Figure 2, a statistical model of a single-hop MI communication system can be developed by defining the source node (Tx) as node 1 and the destination node (Rx) as node \(K\). We wish to evaluate the probabilistic system performance by assuming constant channel conditions over \(r\). One of the fundamental evaluation parameters of any wireless communication system is the average signal-to-noise ratio (average SNR), defined as the statistical average of the received signal over the probability density function (PDF) of fading. Thus, the average error rate performance can be evaluated by averaging the instantaneous error rate over the SNR distribution [35]. For a single hop \(h\) from Tx to Rx, the SNR \(\gamma\) for a single bit is given by  
\[\gamma=\frac{E_{b}}{N_{0}}, \tag{6}\]  
where \(E_{b}\) is the mean energy per transmitted bit and \(N_{0}\) is the power spectral density of thermal noise. Since the underground channel is subject to fading, the received signal power is attenuated by the fading amplitude, denoted by \(\alpha\), which is a random variable whose PDF is conditioned on the nature of fading. The mean square value of the fading amplitude \(\overline{\alpha^{2}}\) is used to define the average SNR per bit, \(\overline{\gamma}\), given by  
\[\overline{\gamma}=\alpha^{2}\frac{E_{b}}{N_{0}}. \tag{7}\]  
The SNR, measured in dB, at the receiver can be defined as  
\[SNR=P_{t}-PL_{MI}-P_{n}, \tag{8}\]  
where \(P_{t}\) and \(P_{n}\) denote the transmit power and noise power in dBm, respectively. A generalised form of the path loss \(PL_{MI}\) in dBm can be written as [14].  
\[PL_{MI}=-10\log\frac{P_{r}}{P_{t}}. \tag{9}\]  
After the SNR (dB) is evaluated from (8), it is first converted to a power ratio in linear scale as:  
\[SNR_{linear}=10^{(SNR/10)} \tag{10}\]  
System performance metrics such as the channel capacity and bit error rate (BER) can be obtained using (10). The channel capacity \(C\) for system bandwidth \(W\) can be obtained using the Shannon capacity theorem:  
\[C=W\ log_{2}(1+SNR_{linear}). \tag{11}\]Further, we compare the BER performance of two modulation schemes, namely binary amplitude shift keying (2 ASK) [18] and binary phase shift keying (2 PSK) [36], by using the following expressions:  
\[BER_{2 ASK}=\frac{1}{2}erfc\sqrt{\frac{SNR_{linear}}{4}}. \tag{12}\]  
\[BER_{2 PSK}=\frac{1}{2}erfc\sqrt{SNR_{linear}}. \tag{13}\]  
The BER is an important statistical performance measure for a channel as it is used to design error control schemes. This statistic can be used to determine the channel behaviour and therefore has vital significance in varying channel conditions. A target symbol error rate (SER) of \(\approx\)10\({}^{-3}\) has been used in [29] for MI-WUSN deployed in underground monitoring applications for varying CSI. Since an underground channel has a dynamic state, this research aims to evaluate the variation of a multihop communication system's performance in terms of the SNR, channel capacity and BER, for a dynamic channel state between hops, in contrast to the commonly used research assumption of an identical channel state between hops. The average bit error probability (BEP) \(E_{h}\) over a single hop \(h\) can be defined as  
\[E_{h}=E_{k}(\gamma_{k}), \tag{14}\]  
where \(E_{k}(\gamma)\) is the average BEP at the receiver node \(k\) in the underground channel. For H hops such that \(h\in H\), we devise a realistic end-to-end performance evaluation strategy in the next section. To extend the single-hop system to a multihop communication model, we considered the following two scenarios.  
1. A generic multihop scenario where all hops are statistically independent and identical
2. A multihop scenario where hops are statistically independent but not identical, which is practical for modeling underground communication