# Ultra-Reliable Link Adaptation for Downlink MISO Transmission in 5G Cellular Networks
## 3 Sources of SINR Variability
### Changes in Interfering Channels  
In addition to changes in the wanted signal, mobility also causes changes in the interfering signals. To obtain the statistics of the SINR, the distribution of total interference power is essential. We consider the interference experienced at the receiver of interest, produced by \(J\) independent interference sources. Here, we assume that there is no change in the interference precoder from the time of measurement to the time of transmission. The realized interference power then is  
\[I_{\tau}=\sum_{j=1}^{J}|\mathbf{h}_{\tau,j}\mathbf{v}_{0,j}|^{2} \tag{8}\]  
where the realized channel \(\mathbf{h}_{\tau,j}\) follows the autocorrelation model Equation (4). Note that the transmission power level is assumed to be absorbed in the channels. Then the total interference power can be represented as a linear combination of magnitude squared of \(J\) mutually independent complex Gaussian RVs with non-zero mean and unit variance, \(I_{\tau}=\sum_{j=1}^{J}\alpha_{j}|X_{j}|^{2}\) where \(\alpha_{j}=(1-\rho^{2})\frac{I_{\rm avej}}{2}\). The RV \(X_{j}\) is again characterized by complex Gaussian distribution with mean \(\frac{\rho}{\sqrt{1-\rho^{2}}}\sqrt{\frac{2I_{0,j}}{I_{\rm avej}}}\) where \(I_{0,j}=|\mathbf{h}_{0,j}\mathbf{v}_{0,j}|^{2}\). The real and imaginary components of RV \(X_{j}\) are Gaussian with unit variance and mean \(\mu_{\Re}=\frac{\rho}{\sqrt{1-\rho^{2}}}\frac{\Re\{\mathbf{h}_{0,j}\mathbf{v} _{0,j}\}}{\sqrt{I_{\rm avej}/2}}\) and \(\mu_{\Im}=\frac{\rho}{\sqrt{1-\rho^{2}}}\frac{\Im\{\mathbf{h}_{j,0}\mathbf{v }_{0,j}\}}{\sqrt{I_{\rm avej}/2}}\), respectively. Hence, \(|X_{j}|^{2}\) can be modeled as a non-central chi square distribution with two degrees of freedom, and non-centrality \(\delta_{j}=\mu_{\Re}^{2}+\mu_{\Im}^{2}\). In the subsequent analysis, we treat this as the sum of two real RVs. Without loss of generality, we can consider these two to have mean \(\sqrt{\delta_{j}/2}\), not \(\mu_{\Re}\) and \(\mu_{\Im}\), _i.e._, we distribute the mean evenly across the real and imaginary components of \(X_{j}\). This is done for ease of notation and analysis, and is precise due to the underlying circular symmetry. Figure 2: Backoff required for URC link adaptation against wanted signal variation. **Left:** Backoff _vs._\(S_{0}/S_{\rm ave}\) for three different \(\rho\) values; **Right:** Backoff _vs._\(\rho\) for three different \(S_{0}/S_{\rm ave}\) values. To characterize the distribution of total interference, we use an expansion discussed in [25]. The probability density function of \(\sum_{i=1}^{n}\alpha_{i}\left(Z_{i}+\delta_{i}\right)^{2}\) where \(Z_{1},Z_{2},...,Z_{n}\) are mutually independent standard normal random variables given by  
\[f_{n}(\mathbf{\alpha},\delta,y)=\sum_{k=0}^{\infty}a_{k}\beta^{-1} \chi^{2}\left(\frac{y}{\beta};n+2k;\Delta\right) \tag{9}\]  
where \(\chi^{2}(y/\beta;n+2k;\Delta)\) is a non-central chi-square distribution, with \(n+2k\) DoF and non-centrality parameter \(\Delta=\sum_{i=1}^{n}\delta_{i}^{2}\). The parameter \(\beta>0\) can be chosen at will, to guarantee convergence. Applying Equation (9) to the distribution of the total interference, the PDF can be expressed as  
\[f_{I}(i_{\tau})=\sum_{n=0}^{\infty}a_{n}\beta^{-1}\chi^{2}\left( \frac{i_{\tau}}{\beta};2(J+n),\Delta\right). \tag{10}\]  
Here, the non-centrality is \(\Delta=\sum_{j=1}^{J}\frac{2\rho^{2}}{(1-\rho^{2})}\frac{I_{0,j}}{I_{\text{ave },j}}\), where \(I_{0,j}=|\mathbf{h}_{0,j}\mathbf{v}_{0,j}|^{2}\) is the measured interference power of the \(j\)th interferer, and the average interference power of interferer \(j\) is \(I_{\text{ave},j}\). The multiplicative coefficients \(a_{n}\) can be derived from a recurrence formula in [26], and \(\beta>0\) can be appropriately selected for fast convergence.