# Ultra-Reliable Link Adaptation for Downlink MISO Transmission in 5G Cellular Networks
## 3 Sources of SINR Variability
### Changes in Wanted Signal Power  
The time selectivity of the desired channel induces SINR variability at the receiver, and is discussed next. For this, we first assume that the interference experienced by the user of interest remains static until the transmission is realized. The distribution of the signal power \(S=|\mathbf{h}_{\rm T}\mathbf{v}_{0}|^{2}\) is thus of interest. From Equation (4) we find  
\[S=\left|\rho\mathbf{h}_{0}\mathbf{v}_{0}+\sqrt{1-\rho^{2}}\mathbf{\hat{h}} \mathbf{v}_{0}\right|^{2}. \tag{5}\]  
Then, for a given channel measurement \(\mathbf{h}_{0}\) and the precoder \(\mathbf{v}_{0}\), \(S\) can be modeled as a magnitude of a sum of two squared independent Gaussian RVs with equal variance \(\alpha=(1-\rho^{2})\frac{S_{\rm ave}}{2}\), and means \(\mu_{\rm R}=\rho\Re\{\mathbf{h}_{0}\mathbf{v}_{0}\}\) and \(\mu_{\rm\Delta}=\rho\Im\{\mathbf{h}_{0}\mathbf{v}_{0}\}\). Note that the probability distribution of the inner product of a channel vector \(\mathbf{h}\) with i.i.d. Gaussian elements and an arbitrary unitary vector \(\mathbf{v}\), is equal to the distribution of an element of the channel vector \(\mathbf{h}\). Therefore, the PDF of the signal power \(f_{S}(s)\) can be characterized by a non-central chi-square distribution with two DoF, and non-centrality \(\delta=(\mu_{\rm R}^{2}+\mu_{\rm\Delta}^{2})/\alpha=\frac{2\rho^{2}}{(1-\rho^{ 2})}\frac{S_{0}}{S_{\rm ave}}\). Here, \(S_{0}=|\mathbf{h}_{0}\mathbf{v}_{0}|^{2}\) is the measured signal power and \(S_{\rm ave}\) is the average signal power at measurement. Note the underlying assumption that both \(S_{0}\) and \(S_{\rm ave}\) are known, along with the signal statistics. This means that the current fading state of the signal is precisely known. The PDF of the signal power is given by  
\[f_{S}(s\mid\mathbf{h}_{0},\mathbf{v}_{0},S_{\rm ave})=\frac{1}{2\alpha}{\rm Exp }\left\{-\frac{s/\alpha+\delta}{2}\right\}\,I_{0}\left(\sqrt{\frac{\delta s}{ \alpha}}\right) \tag{6}\]where \(I_{0}(\cdot)\) is the modified Bessel function of the first kind. The corresponding CDF is given by \(F_{S}(s\mid\mathbf{h}_{0},\mathbf{v}_{0},S_{\mathrm{ave}})=1-Q_{1}(\sqrt{\delta}, \sqrt{s/\pi})\), where \(Q_{M}(a,b)\) is the Marcum-Q function, here with \(M=1\)[24]. The SINR variability caused by changing desired transmission signal power can be visualized from the CDF. The realized SINR is \(\gamma_{\tau}=\frac{S}{b+N_{0}}\), where \(I_{0}=\sum_{j=1}^{J}|\mathbf{h}_{0,j}\mathbf{v}_{0,j}|^{2}\) is the measured interference power which is considered fixed for the given transmission realization. Now, the outage probability becomes \(\mathrm{P}\left(\gamma_{\tau}<\gamma_{t}\right)\). The URC outage probability target is achieved at a SINR threshold \(\gamma_{t}\), and the required backoff to enable robust link adaptation is \(b=\frac{\gamma_{0}}{\gamma_{t}}\). The outage probability can be further simplified to  
\[\mathrm{P}\left(\frac{S}{I_{0}+N_{0}}<\gamma_{t}\right) =F_{X}\left(\gamma_{t}(I_{0}+N_{0})\right)\] \[=1-Q_{1}\left(\sqrt{\frac{2\rho^{2}}{(1-\rho^{2})}\frac{S_{0}}{S _{\mathrm{ave}}}},\sqrt{\frac{2}{(1-\rho^{2})b}\frac{S_{0}}{S_{\mathrm{ave}}}} \right). \tag{7}\]  
We consider a delay between measurement and transmission of \(\tau=1\) ms, in accordance with the URC latency targets. For a carrier frequency \(f=3.6\) GHz, user velocities \(v\approx 8,7\), and \(5\) km/h would lead to auto-correlation coefficients \(\rho\approx 0.97\), \(0.98\) and \(0.99\), respectively. Figure 1 demonstrates how the time selectivity of a channel affects the realized transmission, for three different situations. In the first, the channel of the desired transmission during the measurement is faded (\(S_{0}/S_{\mathrm{ave}}=-10\) dB), in the second it is slightly faded (\(S_{0}/S_{\mathrm{ave}}=-3\) dB), whereas in the third it is not faded (\(S_{0}/S_{\mathrm{ave}}=0\) dB).