## 2 Transfer Function of Galvanic-Type IBC  
### Transfer Function Frequency Response Based on Quasi-Static Field IBC Model  
Based on our previous work [23], we use two pairs of electrodes to transmit and receive electrical signals across the human body in the galvanic-IBC model. As depicted in Figure 1, the human limb can be represented by a length of \(L\) with concentric cylinder layers containing the skin, fat, muscle and bone tissues with radii (\(r_{4}\), \(r_{3}\), \(r_{2}\), \(r_{1}\)), respectively. The electrical properties of each tissue layer are assumed to be homogeneous and characterized by permittivities (\(\varepsilon_{4}\), \(\varepsilon_{3}\), \(\varepsilon_{2}\), \(\varepsilon_{1}\)) and conductivities (\(\sigma_{4}\), \(\sigma_{3}\), \(\sigma_{2}\), \(\sigma_{1}\)). From [23], as the frequency falls below 1 MHz, the inductive effect, propagation effect and radiation from the skin into air can be neglected, while the capacitance effect cannot be ignored. Applying the quasi-static approximation [24] to the Maxwell's Equations, the governing equation of electric potential in cylindrical coordinate system can be simplified into Equation (1) below:  
\[\
abla\cdot\sigma_{cs}(f)\
abla V\approx 0\ \ \ \ \ s=1...4 \tag{1}\]  
where \(V\) represents the electric potential within the human limb, \(\sigma_{cs}(f)\) is composite conductivity of layer \(s\) at frequency \(f\), and:  
\[\sigma_{cs}(f)=\sigma_{s}(f)+j2\pi f\varepsilon_{rs}(f)\varepsilon_{0} \tag{2}\]  
Note that \(\sigma_{s}(f)\) and \(\varepsilon_{rs}(f)\) are the conductivity and relative permittivity of layer \(s\) at frequency \(f\), respectively. \(\varepsilon_{o}\) is the permittivity of free space. Considering with proper electromagnetic boundary conditions, the electric potential distribution on the surface (\(r_{4}\), \(\phi\), \(z\)) of the human limb will be:  
\[\begin{split}\
u_{4}(f,r_{4},\phi,z)=&\sum_{k=1}^{ \infty}\sum_{m=1}^{\infty}\biggl{[}F_{4 dn}  
where \(I_{n}\) is the modified Bessel function of the first kind of order \(n\), \(K_{n}\) is the modified Bessel function of second kind of order \(n\), \(E_{4 kn}\)(\(f\)), \(F_{4 kn}\)(\(f\)), \(G_{4 kn}\)(\(f\)) and \(H_{4 kn}\)(\(f\)) are the parameters on layer \(s=4\) at frequency \(f\) and can be solved by the method stated in [23].  
Using Equation (3), the transfer function of the galvanic-type IBC can be obtained. We assume that ideal electrodes (the thickness of electrode, the impedance between electrode and skin are negligible) are employed for the transmitter and receiver sites. The corresponding electric potential can then be represented by the value at the center of the electrode, and the output differential potential at the receiver can be found as:  
\[Y(f)=V_{4}\left(f,r_{4},\phi_{rp},z_{rp}\right)-V_{4}\left(f,r_{4},\phi_{m},z _{m}\right) \tag{4}\]  
where \(V_{4}\)(\(f\), \(r_{4}\), \(\phi_{rp}\), \(z_{rp}\)) and \(V_{4}\)(\(f\), \(r_{4}\), \(\phi_{m}\), \(z_{m}\)) are the electric potentials at frequency \(f\) at the positive and negative electrode of receiver with locations of (\(r_{4}\), \(\phi_{rp}\), \(z_{rp}\)) and (\(r_{4}\), \(\phi_{m}\), \(z_{m}\)) respectively. Similarly, the applied differential signal at the transmitter side of galvanic-type IBC channel can be expressed as:  
\[X(f)=V_{4}\left(f,r_{4},\phi_{rp},z_{rp}\right)-V_{4}\left(f,r_{4},\phi_{m},z _{m}\right) \tag{5}\]  
where \(V_{4}\)(\(f\), \(r_{4}\), \(\phi_{rp}\), \(z_{rp}\)) and \(V_{4}\)(\(f\), \(r_{4}\), \(\phi_{m}\), \(z_{m}\)) are the electric potentials at the positive and negative electrode of transmitter site, respectively. Using Equations (4) and (5), the transfer function of the Galvanic-type IBC can be found as:  
\[H\left(f\right)=\frac{V_{4}\left(f,r_{4},\phi_{rp},z_{rp}\right)-V_{4}\left(f, r_{4},\phi_{m},z_{m}\right)}{V_{4}\left(f,r_{4},\phi_{p},z_{rp}\right)-V_{4} \left(f,r_{4},\phi_{m},z_{m}\right)}\quad 1 Hz\leq f\leq 1 MHz \tag{6}\]  
Equation (6) is a complex function of both electrical properties and geometry of tissue layers.