## 4 Multi-Site Micro-Scale Diversity Scheme
### Multi-Dimensional Copulas Based Synthesizer  
In this subsection a new multi-dimensional rain attenuation synthesizer is presented, extending the methodology presented in [18], accounting for multiple links used in diversity and multi-hop scenarios. The copula functions are used to couple the single cumulative distributions and to ultimately calculate the joint cumulative distribution. Given that the Cumulative Distribution Functions (CDFs) of dependent random variables are denoted as \(u_{i},i=1,\ldots,n\), the joint statistics can be calculated through copula functions. The Gaussian copula is given by [35]:  
\[P(\mathbf{u})=\boldsymbol{\Phi}_{R}^{n}(\boldsymbol{\Phi}^{-1}(u_{1}),\ldots, \boldsymbol{\Phi}^{-1}(u_{n})) \tag{9}\]  
where \(P(\mathbf{u})\) is the joint CDF of the dependent random variables, \(\boldsymbol{\Phi}_{R}^{n}\) is the CDF of a multivariate normal distribution with zero mean and correlation matrix given by \(\mathbf{R}\) and \(\boldsymbol{\Phi}^{-1}\) is the inverse CDF of a normal distribution with zero mean and unity variance. In (9), the variables \(u_{i}\), \(i=1,\ldots,n\), refer to the univariate CDFs of rain attenuation for the multiple links in the case of spatially-joint statistics or to the univariate CDFs of rain attenuation for different time delays in the case of joint statistics in the temporal domain. In order to introduce different notations for the spatial and temporal dependence the Copula function, the correlation matrix of the Gaussian copula and the CDFs of rain attenuation for spatial joint statistics are denoted as: \(P_{SP}\), \(\mathbf{R}_{SP}\) and \(u^{l},l=1,\ldots,n\) for \(n\) different links, respectively. For the temporal joint statistics, the Copula function is denoted as \(P_{TD}\), the correlation matrix as \(\mathbf{R}_{TD}\) and the CDF of rain attenuation on different time delays as \(u_{i}\), \(j=1,\ldots,K\), with \(j\) being the different time delays. Therefore, the CDF of rain attenuation at link \(l\) and at time instance \(j\) is denoted as \(\mathbf{u}_{j}^{l}\). Equation (9) can therefore be used to describe the joint statistics of rain attenuation in the spatial and the temporal domain using different correlation matrices. In [36], an expression for the calculation of the elements of the correlation matrix \(\mathbf{R}_{TD}\) in order to generate time series is given; in [34] an expression for the calculation of the elements of the correlation matrix \(\mathbf{R}_{SP}\) is given for multiple spatially separated links. It should be noted, however, that the expression derived in [34] is based on measurements with a separation distance of 1.7 km and higher. In Figure 10, the joint CCDF calculated from the emulated time series as explained in Section III is shown versus the one obtained using the Gaussian copulas; the copula parameter used was equal to 0.971. The fitted Gaussian copula exhibits very good performance in modeling the joint statistics for this small separation distance. The copula parameter as proposed in [34] for the same distance, i.e., 386 m, gives a value of 0.9928, however, as already mentioned, the proposed parameter in [34] is derived using data from site diversity experiments with separation distance higher than 1.7 km. Therefore, in our case the copula parameter is extended using a linear relationship between the copula parameter and the distance as shown below for distances between 386 m and 1.7 km:  
\[\rho_{SP}=-0.214d+1.054 \tag{10}\]  
where \(0.386\leq d\leq 1.7\) and \(d\) is in km. Moreover, the proposed expression in (10) has been used to reproduce the experimental data in short scale site diversity in [19]. Therefore it is reasonable that a global relationship for the correlation coefficient for small distances may be derived based on Equation (10) with small refinements obtained through experimental data.