# Ultra-Reliable Link Adaptation for Downlink MISO Transmission in 5G Cellular Networks
## 3 Sources of SINR Variability
### Combined Effect  
To understand the effect of wanted signal and interference variability, one has to combine these into one. The realized SINR Equation (2) can be modeled by \(\gamma_{\tau}=\frac{S}{I+N_{0}}\) where \(S\) and \(I\) are RVs representing the power of the wanted signal and the interference, respectively. Then, the outage probability \(\mathrm{P}\left(\gamma_{\tau}\leq\gamma_{t}\right)\) for SINR with combined wanted signal and interference variability is  
\[\mathrm{P}\left(\gamma_{\tau}<\gamma_{t}\right)=\int_{0}^{\infty}F_{\mathrm{S} }(\gamma_{t}(y+N_{0}))f_{I}(i)di. \tag{18}\]  
The closed form expression for the outage probability can be derived after some mathematical manipulations. It is given by  
\[\mathrm{P}(\gamma_{\tau}<\gamma_{t})=1-\frac{e^{-\left(\frac{\partial}{2}+ \frac{\gamma_{t}N_{0}}{2\pi}\right)}}{2^{p+2q+m+j+n}}\sum_{n=0}^{\infty}\sum_{p =0}^{\infty}\sum_{q=0}^{\infty}\frac{a_{n}\Delta^{q}}{\alpha^{p}\beta^{q+j+n} }\frac{N_{0}^{p+q+j+n}}{(p!)^{2}q!}\left(p!-p\Gamma\left(p,\frac{\delta}{2} \right)\right)U(a,b,\bar{z}) \tag{19}\]  
Figure 6: Effect of interferer precoder variation in multiantenna system. **Left:** PDF of realized interference power due to precoder variation for \(J=3\) interferers; **Right:** CDF of realized interference power. Dash-dot line-\(J=1\), dashed line-\(J=2\) and solid line-\(J=3\). Figure 7: Effect of interferer precoder variation in multi-antenna system. **Left:** CDF of realized interference powers, relative to average interference, \(J=3\) interferers; **Right:** Backoff. Dash-dot line-\(J=1\), dashed line-\(J=2\) and solid line-\(J=3\). where \(U(a,b,\bar{z})\) is a confluent hypergeometric function (A confluent hypergeometric function is defined as \(U(a,b,z)=\frac{1}{\Gamma(a)}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1},\ \ \Re\{a\}>0\)). with \(a=q+J+n\), \(b=p+a+1\), and \(\bar{z}=\left(\frac{\gamma_{l}}{2\kappa}+\frac{1}{2\bar{\rho}}\right)N_{0}\). Here, \(\Gamma(a,b)\) is the incomplete Gamma function. Based on this, a suitable backoff can be numerically evaluated. For the detailed derivation, refer Appendix B. The required backoffs due to mobility for \(J=3\) and \(\rho=0.98\) in four different scenarios are reported in Table 1. The backoffs in the case that only the wanted signal is fading and in the case that only interference is fading are compared to the backoff for the combined effect when both the wanted signal and interference is undergoes fading. It is assumed that noise is negligible, we have \(I_{0}/N_{0}=20\) dB. The average signal-to-interference ratio is \(S_{0}/I_{0}=10\). We see that the larger of the two fading effects dominates the combined effect. The most dramatic effects are caused by the fading of the wanted signal. SINR variability caused by changes in precoders may be considered together with mobility-induced variability. The precoders used on interfering base stations may vary during the transmission. The effective interference is \(I=\sum_{j=1}^{J}|\rho\mathbf{h}_{\tau,j}\mathbf{v}_{\tau,j}+\sqrt{1-\rho^{2}} \tilde{\mathbf{h}}_{j}\mathbf{v}_{\tau,j}|\). For a given precoder \(\mathbf{v}_{\tau,j}\), an individual interference can be modeled as a non-central chi-square distributed variable with PDF \(f_{I}(i_{j}\mid\mathbf{v}_{\tau,j})=\chi^{2}(i_{j}/\alpha_{j};2;\delta_{j})\) where \(\alpha_{j}=(1-\rho^{2})I_{\mathrm{avej}}/2\), and the non-centrality is \(\delta_{j}=\frac{2\lambda_{j}\rho^{2}}{(1-\rho^{2})}\frac{|\mathbf{v}_{I}^{H} \mathbf{v}_{\tau,j}|^{2}}{I_{\mathrm{avej}}}\). The unconditional PDF can be obtained by integrating over the distribution of the inner product \(|\mathbf{v}_{j}^{H}\mathbf{v}_{\tau,j}|^{2}\). If we have \(N_{T}=2\) transmit antennas, this is a uniform distribution, and the integration can be performed in closed form, resulting in \(f_{I}(i_{j})=1-Q_{1}\left(\sqrt{i_{j}},\sqrt{\frac{2\lambda_{j}\rho^{2}}{(1- \rho^{2})I_{\mathrm{avej}}}}\right)\).