# Patient Mobility Support for Indoor Non-Directed Optical Body Area Networks
## 3 Channel Model  
For the optical data transmission, we evaluated the optical wireless channel behavior with respect to the transmitter and the receiver. For indoor applications, the channel can be modeled as a linear time-invariant (LTI) system [16]. In non-directed optical links, the received optical power consists of the power from the directed and non-directed optical links over the surface environment. The received signal thus depends on the incident optical power and the PD responsivity. Hence, the received signal can be written as:  
\[Y=Sx(t)*h(t)+n(t) \tag{1}\]  
where \(S\) is the PD responsivity, \(x(t)\) is the transmitted signal, \(h(t)\) is the channel impulse response, \(n(t)\) is the additive white Gaussian noise (AWGN) generated from the ambient light sources, and \(*\) denotes the convolution [16]. The channel path loss is given as:  
\[H(0)=\int\limits_{-\infty}^{\infty}h(t)dt \tag{2}\]  
Figure 3: Mobility support with the coordinator placed on the body and the receiver on the ceiling. (**a**) Coordinator at shoulder. (**b**) Coordinator at wrist. (**c**) Coordinator at both shoulder and wrist. where \(h(t)\) is the channel impulse response. The reflection coefficient of the wall surface is set to \(\rho\) = 0.8, considering a wavelength of 875 nm [16]. The receiver FOV along with the transmitter and receiver orientation was also considered in analyzing the channel response. With the transmitter mobility, the channel static gain \(H(0)\) randomly varies due to the random transmitter position. To determine the optical gain probability density function (PDF), the experiment was conducted with different transmitter positions distributed uniformly over a 2-dimensional plane in the room. The source was supposed to be fixed at a height of 1 m above the floor. From the simulations considering a uniform distribution of (x, y) transmitter coordinates in the room, we estimated the PDF of the static channel gain. The received power is given as:  
\[P_{R}=H(0)P_{t} \tag{3}\]  
where \(P_{t}\) is the average transmitted optical power. Using Equation (3), the received power at varying distances was measured and the indoor channel path loss characteristics was determined. Based on the above observations, we took into account distance and angle for multiple reflection points in a non-directed optical link. In room dimensions of 5 m \(\times\) 5 m \(\times\) 3 m, each reflection point has a reflection area \(\Delta A\) with an approximate distance between two reflection point sources being 1 cm. Using the illustration shown in Figure 4, we could derive the indoor channel response for multiple reflections. For diffused reflections, the optical channel can be described by two components: (a) Each element of the surface with the area of \(\Delta A\) considered as the receiver and (b) then each element considered as a point source that re-emits the light scaled by reflectivity \(\rho\). The indoor channel path loss for non-directed optical links for multiple reflections from a single wall can be expressed as:  
\[H_{NLOS}\left(T_{x},R_{x},\rho\right)=\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{ N}\frac{\left(m_{i}+1\right)\rho\left(n,m\right)A_{n}\Delta A}{2\pi\left[A_{T_{x},\rho\left(n,m\right)}\right]^{2}\left[A_{R_{x},\rho\left(n,m\right)}\right] ^{2}}\cos\left(\phi_{T_{x}}\left(n,m\right)\right)\cos\left(\psi_{R_{x}}\left( n,m\right)\right) \tag{4}\]  
where \(T_{x}\) is the transmitter, \(R_{x}\) is the receiver, \(m_{l}\) is the Lambertian order, and \(A_{r}\) is the area of the receiver. Each wall was divided into \(N\times M\) reflection points where \(N\) and \(M\) are the number of divisions on the Y- and Z- axis. \(\rho\) represents the reflective property of the wall surface. Each point on the wall surface has a different distance and angle between the transmitter and receiver. The propagation distance between the transmitter and receiver is expressed as:  
\[l_{x}(n)=\sqrt{\left(n-\frac{1}{2}\right)^{2}+\left(\frac{l_{x}}{2}\right)^{2}} \tag{5}\]  
Figure 4: Analysis of the multiple reflected signals.