# Ultra-Reliable Link Adaptation for Downlink MISO Transmission in 5G Cellular Networks
## 3 Sources of SINR Variability
### Changes in Precoders of Interferers  
The precoders used in interfering base stations may not be the same as the ones used when the channel is measured. This may cause changes in the measured interference power, causing CSI mispredictions and SINR variability [8]. This effect is particularly strong in multiple input single output (MISO) channels. We assume that the precoder of the desired transmission is selected based on the wanted channel \(\mathbf{h}_{i}\), and the precoders of \(J\) interfering base stations are selected according to the wanted channels of the intended receivers of the interfering transmissions. The precoders \(\mathbf{v}_{j,\tau}\) used at the interfering base stations depend on the scheduled receivers in the interfering cells. We  
Figure 5: Required backoff for three different \(I_{0}/I_{\mathrm{ave}}\). Three curves per \(I_{0}/I_{\mathrm{ave}}\) represents \(J=1,2,3\). The backoff for \(J=1\) is always largest and for \(J=3\) smallest for a given \(I_{0}/I_{\mathrm{ave}}\). assume that these are uniformly selected from the space of all MISO precoders. If there is no mobility, the received SINR for the desired transmission is  
\[\gamma_{\tau}=\frac{\lambda_{0}}{\sum_{j=1}^{J}\lambda_{j}|\mathbf{v}_{j}^{H} \mathbf{v}_{\tau,j}|^{2}+N_{0}}. \tag{15}\]  
Here, \(\lambda_{0}=||\mathbf{h}_{0}||^{2}\) is the wanted signal power. The powers of interfering channels are \(\lambda_{j}\) correspond to the interfering channels \(\mathbf{h}_{\tau,j}=\mathbf{h}_{0,j}\) in Equation (3), and \(\mathbf{v}_{j}=h_{j}/\sqrt{\lambda_{j}}\) are the normalized interfering channels. We assume that the receiver is able to measure the average interference. The measured SINR is thus \(\gamma_{0}=\frac{\lambda_{0}}{I_{0}+N_{0}}\), where \(I_{0}=\mathrm{E}_{\mathbf{v}_{\tau,j}}\{\sum_{j=1}^{J}\lambda_{j}|\mathbf{v}_ {j}^{H}\mathbf{v}_{\tau,j}|^{2}\}\). For a given channel vector distributed on a complex sphere with radius \(\lambda_{j}=||\mathbf{h}_{0,j}||^{2}\), the cumulative distribution of an inner product of the channel vector and a unitary precoder \(\mathbf{v}_{\tau,j}\), \(Z_{j}=\lambda_{j}|\mathbf{v}_{j}^{H}\mathbf{v}_{\tau,j}|^{2}\) can be obtained from [29]  
\[F_{Z}(z_{j})=1-\left(1-\frac{z_{j}}{\lambda_{j}}\right)^{N_{T}-1}. \tag{16}\]  
The corresponding PDF is then \(f_{Z}(z_{j})=\left(\frac{N_{T}-1}{\lambda_{j}}\right)\left(1-\frac{z_{j}}{ \lambda_{j}}\right)^{N_{T}-2}\). Hence, the PDF of the total interference power \(Y=\sum_{j=1}^{J}Z_{j}\) induced from \(J\) independent interferes can be written as  
\[f_{Y}(y)=\left(\frac{N_{T}-1}{\lambda_{1}}\right)\left(1-\frac{z}{\lambda_{1} }\right)^{N_{T}-2}*\ldots*\left(\frac{N_{T}-1}{\lambda_{J}}\right)\left(1- \frac{z}{\lambda_{J}}\right)^{N_{T}-2} \tag{17}\]  
where \(*\) is the convolution operation. Hence, the total interference power is distributed over \([0\ \ \sum_{j}\lambda_{j}]\). This is a convolution of functions with compact support. Accordingly, for \(J\) interferers, the CDF is generically divided into \(2^{l}\) disjoint regions with different functional form. The boundaries of the regions depend on the relative interference powers, see e.g. [30]. As an example of interference variation due to changes of precoders in neighboring cells, we consider a system with \(J=3,2\), or \(1\) interferers. The eigenvalues of the interferers are \(\lambda_{1}=\lambda_{2}=\lambda_{3}=1\) for \(J=3\), \(\lambda_{1}=2\) and \(\lambda_{2}=1\) for \(J=2\), and \(\lambda_{1}=3\) for \(J=1\). We consider four different numbers of transmit antennas at the interferers, \(N_{T}\in\{2,5,10,100\}\). Figure 6 shows the PDF and CDF of the interference at the time of transmission, when the interferers select precoders randomly. For clarity of the figures, \(N_{T}=100\) is omitted. For \(N_{T}=2\), the \(10^{-5}\) point of the complementary CDF of interference is virtually indistinguishable from the worst case interference.