# Study of 5G-NR-MIMO Links in the Presence of an Interferer
## Appendix A
\]  
Let \(\frac{\left|(v_{2}+u_{2})^{*}(v_{1}+u_{1})\right|}{\left\|v_{2}+u_{2}\right\| \cdot\left\|v_{1}+u_{1}\right\|}=cos(\varphi)\) and \(\frac{\left|z_{2}^{*}z_{1}\right|}{\left\|z_{2}\right\|\cdot\left\|z_{1}\right\| }=cos(\psi)\), where \(\varphi,\psi\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Thus, since \(\left\|v_{1}+u_{1}\right\|\cdot\left\|z_{1}\right\|\leq\left\|v_{1}+u_{1} \right\|^{2}+\left\|z_{1}\right\|^{2}\) and \(\left\|v_{2}+u_{2}\right\|\cdot\left\|z_{2}\right\|\leq\left\|v_{2}+u_{2} \right\|^{2}+\left\|z_{2}\right\|^{2}\), it follows that  
\[\frac{\left|\left(v_{2}+e^{i\theta_{2}}w_{2}\right)^{*}\left(v_{ 1}+e^{i\theta_{1}}w_{1}\right)\right|^{2}}{\left\|v_{2}+e^{i\theta_{2}}w_{2} \right\|^{2}\cdot\left\|v_{1}+e^{i\theta_{1}}w_{1}\right\|^{2}}\] \[=\frac{\left|\left(v_{2}+u_{2}\right)^{*}\left(v_{1}+u_{1}\right) \right|^{2}}{\left(\left\|v_{2}+u_{2}\right\|^{2}+\left\|z_{2}\right\|^{2} \right)\cdot\left(\left\|v_{1}+u_{1}\right\|^{2}+\left\|z_{1}\right\|^{2} \right)}\] \[\leq\frac{\left|\left(v_{2}+u_{2}\right)^{*}\left(v_{1}+u_{1} \right)\right|^{2}+2\left|\left(v_{2}+u_{2}\right)^{*}\left(v_{1}+u_{1}\right) \right|\left|z_{2}^{*}z_{1}\right|+\left|z_{2}^{*}z_{1}\right|^{2}}{\left( \left\|v_{2}+u_{2}\right\|^{2}+\left\|z_{2}\right\|^{2}\right)\cdot\left( \left\|v_{1}+u_{1}\right\|^{2}+\left\|z_{1}\right\|^{2}\right)}\] \[\leq\frac{\left|\left(v_{2}+u_{2}\right)^{*}\left(v_{1}+u_{1} \right)\right|^{2}}{\left\|v_{2}+u_{2}\right\|^{2}\cdot\left\|v_{1}+u_{1} \right\|^{2}}+\] \[\quad+2\frac{\left|\left(v_{2}+u_{2}\right)^{*}\left(v_{1}+u_{1} \right)\right|\left|z_{2}^{*}z_{1}\right|}{\left\|v_{2}+u_{2}\right\|^{2}+ \left\|z_{2}\right\|^{2}\right)\cdot\left(\left\|v_{1}+u_{1}\right\|^{2}+ \left\|z_{1}\right\|^{2}\right)}+\] \[\quad+\frac{\left|z_{2}^{*}z_{1}\right|^{2}}{\left\|z_{2}\right\| ^{2}\cdot\left\|z_{2}\right\|^{2}}\] \[\leq\cos^{2}(\varphi)+2\cos(\varphi)\cos(\psi)+\cos^{2}(\psi)\] \[=(\cos(\varphi)+\cos(\psi))^{2}.\]  
Now, assume that \(\varphi,\psi\in\left[-\frac{\pi}{2},-\frac{\pi}{2}+\delta\right]\) or \(\varphi,\psi\in\left[\frac{\pi}{2}-\delta,\frac{\pi}{2}\right]\) for some small \(\delta>0\). Thus, \(\cos(\varphi),\cos(\psi)\leq\sin(\delta)\leq\delta\), implying that \((\cos(\varphi)+\cos(\psi))^{2}\leq 4\delta^{2}\). Therefore, for Equation (A2) to hold, it must be the case that \(4\delta^{2}<\epsilon\), or \(\delta<\frac{\sqrt{\epsilon}}{2}\). **Theorem A1**. : _Let \(0<\epsilon<1\) be the distance from orthogonality that the system allows. In the notions given above, assume that \(\varphi,\psi\) are uniformly distributed in \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\times\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Thus, the probability that the channel response matrix of the signal received vector, combined from the transmitter and the jammer links, has orthogonal columns up to \(\epsilon\), which is less than \(\frac{\epsilon}{\pi^{2}}\)._Proof of Theorem 1.: Let \(\delta<\frac{\sqrt{\varepsilon}}{2}\). Therefore,  
\[p\Bigg{(}\frac{\left|\left(v_{2}+e^{\left|\mathsf{z}_{1,2}}w_{1}+e^{ \left|\mathsf{z}_{2,2}w_{2}\right.\right\rangle}\right)^{*}\left(v_{1}+e^{ \left|\mathsf{z}_{1,1}w_{1}+e^{\left|\mathsf{z}_{2,1}w_{2}\right.\right\rangle }\right)^{2}}}{\left\|v_{2}+e^{\left|\mathsf{z}_{1,2}w_{1}+e^{\left|\mathsf{z}_ {2,2}w_{2}\right.\right\|^{2}}}\right\|^{2}\cdot\left\|v_{1}+e^{\left|\mathsf{z }_{1,1}w_{1}+e^{\left|\mathsf{z}_{2,1}w_{2}\right.\right\|^{2}}}\right\|^{2}}< \epsilon\Bigg{)}\] \[\leq P\Big{(}\left(\cos(\varphi)+\cos(\psi)\right)^{2}<\epsilon \Big{)}\] \[=\frac{\left|\left(v_{2}+u_{2}\right)^{*}\left(v_{1}+u_{1}\right) \right|^{2}}{\left(\left\|v_{2}+u_{2}\right\|^{2}+\left\|z_{2}\right\|^{2} \right)\cdot\left(\left\|v_{1}+u_{1}\right\|^{2}+\left\|z_{1}\right\|^{2} \right)}\]0  
In order to illustrate the result of the theorem, assume that the interference is acting by exchanging the UE channel-matrix columns, i.e., \(e^{\left|\mathsf{z}_{1,2}w_{1}+e^{\left|\mathsf{z}_{2,2}w_{2}\right.