# Realistic Simulation for Body Area and Body-To-Body Networks
## 3 Proposed Simulator
### Semi-Deterministic (Stochastic) Radio Link and Mobility Modeling
#### 3.1.3 On-Body Channel Model  
On-body channels are classically investigated on a scenario-based approach [38], as body shadowing and human mobility are the factors that have an impact to the greatest extent on their characteristics. In this sense, a scenario is the combination of the specific Tx and Rx antenna locations on the body and the human body mobility condition. The implementation of the on-body channel model is rather simple and mostly relies on the generation of time-correlated long- and short-term fading with given distribution parameters. Moreover, as long-term fading of on-body channels strongly depends on body shadowing, it is relevant to investigate their inter-link correlation, which can significantly affect, for instance, wrist-related links, where the arm movement effects may create correlated links. On the other hand, the average channel gain only depends on the on-body antenna position, as shown in Table 1.  
The temporal correlation of both short- and long-term fading is enforced by applying an auto-regressive (AR) method, which linearly combines outputs at previous times with a temporally-uncorrelated input random process. Its formulation to derive an output \(y\) at a time instant \(t\) is the following:  
\[y\left(t\right)=-\sum_{k=1}^{p}a_{k}y\left(t-k\right)+x\left(t\right) \tag{4}\]  
where \(p\) is the order of the AR model, \(x\) is the input process and \(a_{k}\) are the weights that minimize the variance of prediction error. Table 2 provides the AR coefficients for on-body links. The implementation requires a large number of significant figures to preserve the convergence of the AR method. Long-term fading inter-link correlation is implemented through a Cholesky factorization of the correlation matrix. \[ \begin{bmatrix}x_{c}^{\left(hb\right)}\left(t\right)\\ x_{c}^{\left(hwt\right)}\left(t\right)\\ x_{c}^{\left(bw\right)}\left(t\right)\end{bmatrix} = \begin{bmatrix}x^{\left(hb \right)}\left(t\right)\\ x^{\left(hwt\right)}\left(t\right)\\ x^{\left(bw\right)}\left(t\right)\end{bmatrix} \cdot\mathbf{C} \tag{5}\]  
where \(\mathbf{C}\) is the 3 \(\times\) 3 Cholesky factorized matrix, \(x_{c}\) is the correlated process and the indices of the superscripts identify the different type of channel ( h stands for head, b for belt and w for wrist). \begin{table}
\begin{tabular}{c c} \hline \hline
**On-Body Link** & **Gain (dB)** \\ \hline
**Head-to-Belt** & \(-48.2\) \\
**Head-to-Wrist** & \(-53.9\) \\
**Belt-to-Wrist** & \(-50.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average gain for each on-body link. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**AR Coefficient** & **Belt-to-Wrist** & **Belt-to-Head** & **Head-to-Wrist** \\ \hline \(a_{0}\) & \(-1.9734512929\) & \(-2.0672207748\) & \(-2.0268577284\) \\ \(a_{1}\) & \(0.8052924686\) & \(0.9814623443\) & \(0.9057458068\) \\ \(a_{2}\) & \(0.1558380462\) & \(0.1007066836\) & \(0.1273802520\) \\ \(a_{3}\) & \(0.0297438008\) & \(0.0105872912\) & \(0.0178147987\) \\ \(a_{4}\) & \(0.0047857128\) & \(0.004463117\) & \(0.0012784956\) \\ \(a_{5}\) & \(-0.0004256897\) & \(-0.0013910969\) & \(-0.0018884276\) \\ \(a_{6}\) & \(-0.0010198892\) & \(-0.0021916391\) & \(-0.0026983122\) \\ \(a_{7}\) & \(0.0029269547\) & \(-0.0022352955\) & \(-0.0015016341\) \\ \(a_{8}\) & \(0.0258211963\) & \(0.0030632551\) & \(0.0118456028\) \\ \(a_{9}\) & \(-0.0494859747\) & \(-0.0231942330\) & \(-0.0310823183\) \\ \(\sigma_{E}\) & \(7.651762983\times 10^{-7}\) & \(9.260825677\times 10^{-7}\) & \(9.281826210\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Table of auto-regressive model weights for on-body channels. In our implementation, the Cholesky factorized matrix is:  
\[\mathbf{C}= \begin{bmatrix}1&0.21&0.51\\ 0&0.9777&0.228\\ 0&0&0.8294\end{bmatrix}