## 2 Transfer Function of Galvanic-Type IBC
### Transfer Function with Correction Factor  
Even though both the calculated and measured results of the IBC channel exhibit similar high pass characteristics, the calculated values from Equation (6) obviously deviate somewhat from the measurements. These sources of error are intrinsic and can result from several aspects. The first obvious reason for deviations comes from the fact that the actual shapes of all subjects' limbs vary somewhat from our concentric cylindrical model. Another major reason is due to the uncertainties of electrical properties in our _in vivo_ subjects' tissues. In the galvanic-type IBC, the signal mainly flows within the human body, especially with low frequency carrier. The electrical properties of the tissues significantly influence the propagation of the signal. From the literatures, the physiological status variations of the tissues induce fluctuations in electrical properties [30]. Additionally, in accordance to the experiment procedures of [PERSON] _et al._[31], the measurements have \(\pm\)(15-25)% uncertainties at low frequencies due to physiological processes and there were three sources of materials, including human autopsy materials and sheep [31]. From the top of Figure 3, one can observe that obvious variations happen at low frequencies (\(<\)1 kHz), which is caused by aforementioned effect and/or other nearby artifacts there. Owing to the larger attenuation and fluctuation there, this frequency band would not be used in IBC signal transmission. Hence, we will focus our investigation between 1 kHz to 1 MHz for the rest of this paper. In order to take into account of non-avoidable variations, a correction factor \(A(f)\) is introduced to Equation (6) in order to compensate the variation effects. Hence, the transfer function of the IBC would become:  
\[H_{\Pi}(f)=A\big{(}f\big{)}.\frac{V_{\pm}\big{(}f,r_{4},\phi_{p},z_{\mp}\big{)} -V_{4}\big{(}f,r_{4},\phi_{m},z_{m}\big{)}}{V_{4}\big{(}f,r_{4},\phi_{p},z_{m} \big{)}-V_{4}\big{(}f,r_{4},\phi_{m},z_{m}\big{)}} \tag{7}\]  
where \(A(f)\) is the semi-empirical correction factor, related to the individual human body. Note \(A(f)\) corrects the inadequacies of the uncompensated model that did not account for individual subject characteristic. As seen in Figure 3, both the calculation and measurement results possess the similar trend, their difference is approximately constant. This is especially true in our frequency of interest (_i.e._, 20 kHz-1 MHz), for which both experimental and calculation results are relatively flat. Hence, for the ease of handling at this stage to determine the correction factor, we assumed the value of \(A(f)\) is independent of frequency \(f\) (denoted as \(A\)). Alternatively, the transfer function in dB is given as:  
\[H_{\Pi dB}(f)=20\cdot\log_{10}\big{(}H(f)\big{)}+K \tag{8}\]  
where \(K=20\log_{10}(A/)\). The determination of correction factor \(A\) in Equation (7) is equivalent to determining the parameter \(K\) in Equation (8). Note that \(K\) is subject-dependent to compensate the variations of tissues' parameters in different subjects. In order to find K numerically, we replace \(K\) with variable \(k\), and have:  
\[H_{\Pi dB}\big{(}f,k\big{)}=20\cdot\log_{10}\big{(}H\big{(}f\big{)}\big{)}+k \tag{9}\]  
Then \(K\) can be found by minimizing the mean square error (MMSE) between the calculated result in Equation (9) and the measurements over the less fluctuation frequency band (1 kHz to 1 MHz):\[K=\underset{k}{\arg\min}\;E\Bigg{[}\bigg{(}\big{|}\hat{H}_{dB}\big{(}f\big{)}\! \cdot\!H_{\text{T}\text{L}dB}\big{(}f,k\big{\}}\big{)}^{2}\bigg{]} \tag{10}\]  
where \(H_{\text{n}dB}\)(\(f\),\(k\)) represents calculation value for the individual person plus the value \(k\). And \(\hat{H}_{dB}\)(\(f\)) is the measurement value for the subject. With the introduction of the constant correction factor \(K\) in Equation (8), the transfer function is then re-evaluated. By solving Equation (10), \(K\) is found to be around \(+5.5\) dB and \(+1.5\) dB for subject A and subject B, respectively.