## 4 Multi-Site Micro-Scale Diversity Scheme
### Multi-Dimensional Copulas Based Synthesizer
In order to generate rain attenuation time series zero mean and unity variance Gaussian random samples for each of the links are first generated, correlated in the time domain using the correlation matrix \(\mathbf{R}_{TD}\). These samples actually correspond to samples of \(\mathbf{\Phi}^{-1}(\mathbf{u}_{j}^{l})\) which refer to time series with only the temporal correlation having been taken into account. Up to this point, the spatial correlation has not been considered, as these time series are produced independently for each link. Then for every time instance \(j\) the vector \(\mathbf{F}_{j}=[\mathbf{\Phi}^{-1}(\mathbf{u}_{j}^{l}),\dots,\mathbf{\Phi}^{ -1}(\mathbf{u}_{j}^{n}),\dots,\mathbf{\Phi}^{-1}(\mathbf{u}_{j}^{n})]^{T}\) is multiplied on the right side by the Cholesky decomposition of \(\mathbf{R}_{SP}\). It should be noted that the multiplication of \(\mathbf{F}\) by the Cholesky decomposition of \(\mathbf{R}_{SP}\) will again result into Gaussian random variables. Therefore, on every given time instance \(j\) the random variables \(\mathbf{\Phi}^{-1}(\mathbf{u}_{j}^{l})\) are now correlated also in the spatial domain. In order to generate time series of rain attenuation the CDF of \(\mathbf{\Phi}^{-1}(\mathbf{u}_{j}^{l})\) has to be calculated first, considering normal zero mean and unity variance distribution in order to obtain \(\mathbf{u}_{j}^{l}\); then the inverse CDF of rain attenuation distribution is calculated on the values of \(\mathbf{u}_{j}^{l}\). The step-by-step algorithm for generating rain attenuation time series of duration \(T\) with sampling time \(t_{s}\) for \(n\) links is as follows:  
1. Calculate the number of samples \(K\) for the time series corresponding to each link, \(K=T/t_{s}\);
2. Generate \(K\) samples of Gaussian random variables with zero mean and covariance matrix \(\mathbf{R}_{TD}\);
3. Repeat step 2 \(n\) times, equal to the number of links;
4. From the samples created in steps 2 and 3, create the samples of vectors \(\mathbf{F}_{j}\) considering the samples from the different links referring to the same time instance \(j\);
5. Multiply on the right side every sample of the vector \(\mathbf{F}_{j}\) with the Cholesky decomposition of \(\mathbf{R}_{SP}\);
6. Create the matrix \(\mathbf{U}=[\mathbf{F}_{1},\dots,\mathbf{F}_{K}]^{T}\);
7. Calculate on every element the normal CDF;
8. On every element of each line \(i\) calculate the inverse CDF of rain attenuation induced on link \(i\);
9. Every line of the matrix created in step 8 should now correspond to the rain attenuation time series on each link. An example of the time series generated is shown in Figure 11, where three stations are considered to form an equilateral triangle with sides equal to 386 m. The joint-minimum of the rain attenuation time series is also shown in the same figure. To validate the above  
Figure 10: Joint CCDF of rain attenuation calculated from the minimum of the time series using SST vs. the fitted Gaussian copula for 19.701 GHz, i.e., Ka-band downlink. methodology before applying it to the multi-dimensional case, the joint CCDF of rain attenuation as obtained from the emulated data (i.e., using the SST) vs. simulated data using the copula synthesizer for two links is presented in Figure 12; from the results obtained it can be concluded that the synthesized data are in excellent accordance with the emulated ones. The second order statistics (fade durations, fade slopes, cross correlation etc.) are not investigated since they are beyond the scope of this contribution.