## 2 Dynamic Lognormal Shadowing Framework
### Dynamic Model
Thus, existing models for the shadowing process \(\mathbf{S}(t)\) permit the derivation of the covariance matrix \(\mathbf{C}_{\mathbf{X}}\) of \(\mathbf{X}_{t}\). Then, equation (25) can be used for the determination of the transformation matrix \(\mathbf{B}\), that is required for the dynamic modeling of the process \(\mathbf{X}_{t}\) in (16), given the matrix \(\mathbf{A}\) and the stationary covariance matrix \(\mathbf{C}_{\mathbf{X}}\). If we denote:  
\[d\mathbf{X_{t}}=\mathbf{A}\cdot\mathbf{X_{t}}dt+\mathbf{B}\cdot d\mathbf{W_{t} },\,\mathbf{X_{0}}=\mathbf{x}_{0} \tag{16}\]1  
then from (25) we can determine that \(\mathbf{G}\) is given by  
\[d\mathbf{X_{t}}=\mathbf{A}\cdot\mathbf{X_{t}}dt+\mathbf{B}\cdot d\mathbf{W_{t} },\,\mathbf{X_{0}}=\mathbf{x}_{0} \tag{16}\]2  
The matrix \(\mathbf{C}_{\mathbf{X}}\) as a covariance matrix is real and symmetric and from (27) the same is true also for \(\mathbf{G}\). The decomposition of \(\mathbf{G}\) as the product \(\mathbf{B}\mathbf{B}^{T}\) is straightforward and can be realized via Cholesky decomposition. By use of the solution process (18) a straight-forward calculation based on the properties of the stochastic integral leads to the cross-covariance for two time instants \(t_{1}<t_{2}\)  
\[d\mathbf{X_{t}}=\mathbf{A}\cdot\mathbf{X_{t}}dt+\mathbf{B}\cdot d\mathbf{W_{t} },\,\mathbf{X_{0}}=\mathbf{x}_{0} \tag{16}\]3  
in direct relation to (13). The model as formulated corresponds to the case of a fixed wireless system and captures the temporal variability of shadowing due to obstacle movement. Of great interest though is the case of mobile wireless systems. We consider the case of \(m\) base stations and \(l\) mobile users moving along the trajectories \(\mathbf{r}_{j}(t)=(x_{j}(t),y_{j}(t))\), \(j=1\ldots l\). By discretizing each trajectory in \(k_{l}\) points, separated by a distance corresponding to the distance scale of the shadowing process, we result in \(n=m\underset{l}{\sum}k_{l}\) links. The model may be used now for these successive locations providing the temporal evolution of the shadowing experienced by the mobile users. The stationary covariance matrix contains the covariances of any pair of locations for the corresponding signals of any pair of base stations. By adding more locations we may include the trajectories of other mobiles too. However, such an implementation provides more than we need, as the process models the temporal evolution for each point of the trajectories, i.e., for every time instant we get \(n=m\underset{l}{\sum}k_{l}\) shadowing values and we actually need \(ml\). By using (12)-(14) and the fact that \(\mathbf{x}_{2}=\mathbf{x}_{1}+\mathbf{v}(t_{2}-t_{1})\), where \(\mathbf{v}\) is the velocity vector of the mobile, we see that for a distance covered, such that the shadowing process may be considered locally stationary, the covariance of the shadowing experienced by a mobile user is described as:  
\[d\mathbf{X_{t}}=\mathbf{A}\cdot\mathbf{X_{t}}dt+\mathbf{B}\cdot d\mathbf{W_{t} },\,\mathbf{X_{0}}=\mathbf{x}_{0} \tag{16}\]4  
where \(a^{\prime}(\mathbf{x})=a(\mathbf{x})+|\mathbf{v}|/r_{c}\). This approximation may be considered valid as long as the movements are not of large scale and a stationary covariance matrix can approximate the shadowing correlations and cross-correlations, irrespective of the users mobility [37]. In this case the correlations and cross-correlations have to be calculated for a representative separation angle and distance while the dynamic parameters account also for user's mobility. Here we have to note, that the proposed model is based on the multi-dimensional SDE in equation (16) with the appropriate diffusion and drift coefficients. The SDE is not directly related to motion, but provides the necessary spatial and temporal variability through the corresponding covariance matrices and the rest parameters. The covariance matrix that has been assumed has been selected as well accepted and proved in the related literature concerning the spatial variability of shadowing.