# Ultra-Reliable Link Adaptation for Downlink MISO Transmission in 5G Cellular Networks
## 3 Sources of SINR Variability
### Changes in Interfering Channels
In Appendix A, it is argued that a choice guaranteeing convergence is  
\[\beta=\frac{2\alpha_{\text{max}}\alpha_{\text{min}}}{\alpha_{\text{ max}}+\alpha_{\text{min}}} \tag{11}\]  
where \(\alpha_{min}\) and \(\alpha_{max}\) are the minimum and maximum values of \(\{\alpha_{i}\}_{1}^{l}\). In practice, some of the interference signals may be relatively small when compared to some others (\(I_{0,j}/\max\{I_{0,j}\}<0.1\)). This may lead to extremely slow convergence of the series Equation (10), even with an optimal selection of \(\beta\). For details, see Appendix A. To ensure numerical stability, some weak interferers may be approximated. A Gaussian approximation of the interference power distribution is not optimal. For URC operation, we are especially interested in the tail of the total interference distribution, which a Gaussian approximation would not capture. Instead, the large fluctuations of the distribution in (10) can be treated with a moment matching non-central chi-square approximation. The interference powers of some of the weak interferers is added to the strongest interferer, and moment matching is applied to obtain an equivalent distribution model [27]. For the total realized interference in Equation (8), the \(k\)th cumulant is  
\[\kappa_{k}=2^{k-1}(k-1)!\left[\sum_{j=1}^{J}\alpha_{j}^{k}+k\sum_ {j=1}^{J}a_{j}^{k}\delta_{j}\right]. \tag{12}\]  
A non-central chi-square approximation can be derived such that the first two cumulants, the mean \(\kappa_{1}\) and the variance \(\kappa_{2}\) equal the corresponding statistics of the total realized interference power. The non-centrality parameter and the DoF of the chi-square approximation \(\chi^{2}(x;\bar{J};\bar{\Delta})\) can be directly obtained as \(\bar{J}=2\kappa_{1}-\kappa_{2}/2\) and \(\bar{\Delta}=\kappa_{2}/2-\kappa_{1}\). To achieve more accuracy at the tail of the distribution, the first four cumulants may be considered. The parameters \(\bar{\Delta}\) and \(\bar{J}\) are determined so that the skewness (\(\kappa_{3}\)) of the actual distribution and the approximation are equal and the difference between the kurtoses (\(\kappa_{4}\)) of two distributions is minimized [27]. If \(s_{1}^{2}>s_{2}\), the approximation is characterized by \(\bar{\Delta}=s_{1}a^{3}-a^{2}\) and \(\bar{J}=a^{2}-2\bar{\Delta}\), with \(a=1/(s_{1}-\sqrt{s_{1}^{2}-s_{2}})\). Else, if \(s_{1}^{2}\leq s_{2}\), \(\bar{\Delta}=0\) and \(\bar{J}=a^{2}\) with \(a=1/s_{1}\). Here, \(s_{1}=\kappa_{3}/\kappa_{2}^{3/2}\) and \(s_{2}=\kappa_{4}/3\kappa_{2}^{2}\). The method proposed in [27] shows much smaller approximation errors for the model used in Equation (10), compared to [PERSON]'s method [28]. Figure 3 shows the probability distribution of the total interference power, and its moment matching chi-square approximation. The figure on the left side depicts the theoretical probability density of (10) for \(J=3\) independent interference sources together with a simulated Monte Carlo realization of the distribution, and the moment matching approximations. The figure on the right side zooms in to the tail of the distribution with very large interferences. The approximation with the first four cumulants is especially good for the tail of the distribution. It is tight to the improbable very high interferences, that are of interest when addressing URC. Figure 4 shows CDFs of realized total interference powers when there are \(J=1,2,3\) independent interferers. The measured total interference \(I_{0}=\sum_{j}I_{0,j}\) is set to \(0,-3,-10\) dB, as compared to the average total interference \(I_{\text{ave}}=\sum_{j}I_{\text{ave},j}\). We assume that the average interference power is the same for all the individual interferers. When there are \(J=2\) interferers, the measured interference power of the first interferer is assumed to be \(2/3\) of the total interference, whereas the second interferer is measured to contribute \(1/3\) of the total. For \(J=3\) interferers, the ratios of the measured interference powers is assumed to be \(1/2\), \(1/3\), \(1/6\).