# Study of 5G-NR-MIMO Links in the Presence of an Interferer
## Appendix A  
This appendix discusses the mathematical analysis of the interfering effect on the channel matrix orthogonality. Assuming all transmissions signals and interferers are M-PSK modulated, which have equal energy, i.e., CE, we can replace any symbol \(j_{1},j_{2}\) with any other symbol, such as \(s_{1}\) and \(s_{2}\), by multiplying it by \(e^{\beta}\), for \(\theta\in\left\{e^{j\frac{\pi}{M}(1+2n)}\right\}\). We can write  
\[\left[ \begin{array}{c}j_{1}\\ j_{2}\end{array} \right]=\left[ \begin{array}{cc}e^{\beta_{1}}&0\\ 0&e^{\beta_{2}}\end{array} \right]\left[ \begin{array}{c}s_{1}\\ s_{2}\end{array} \right].\]  
We therefore can write Equation (11) as  
\[\vec{R} = \begin{bmatrix}r_{1}(1)\\ r_{1}^{*}(2)\\ r_{2}(1)\\ r_{2}^{*}(2)\end{bmatrix}  = \sqrt{\frac{1}{P}} \begin{pmatrix}h_{11}&h_{12}\\ h_{12}^{*}&-h_{11}^{*}\\ h_{21}&h_{22}\\ h_{22}^{*}&-h_{21}^{*}\end{pmatrix} +\sqrt{\frac{1}{SR}} \begin{bmatrix}\delta_ {11}&\delta_{12}\\ \delta_{12}^{*}&-\delta_{11}^{*}\\ \delta_{21}&\delta_{22}^{*}\\ \delta_{22}^{*}&-\delta_{21}^{*}\end{bmatrix}\begin{bmatrix}e^{\beta_{1}}&0\\ 0&e^{\beta_{2}}\end{bmatrix}\begin{bmatrix}s_{1}\\ s_{2}\end{bmatrix} +\sqrt{\frac{P}{SNR}} \begin{bmatrix}n_{1}(1)\\ n_{1}^{*}(2)\\ n_{2}(1)\\ n_{2}^{*}(2)\end{bmatrix}  \tag{13}\]  
The combined response from the transmitter and jammer, as observed from the \(RX\), can be considered the sum of two matrices acting on the same vector of symbols \(\vec{S}\). Equation (13) presents the channel matrix that includes signal transmission and interference. For simplicity, let  
\[\left[ \begin{array}{cc}v_{1}&v_{2}\end{array} \right]= \begin{bmatrix}h_{11} &h_{12}\\ h_{12}^{*}&-h_{11}^{*}\\ h_{21}&h_{22}\\ h_{22}^{*}&-h_{21}^{*}\end{bmatrix} \]  
denote the channel matrix of the UE, and let  
\[\left[ \begin{array}{cc}w_{1}&w_{2}\end{array} \right]=\frac{1}{\sqrt{SR}} \begin{bmatrix}\delta_{11}&\delta_{12}\\ \delta_{12}^{*}&-\delta_{11}^{*}\\ \delta_{21}&\delta_{22}\\ \delta_{22}^{*}&-\delta_{21}^{*}\end{bmatrix} \]  
denote the channel matrix of the interference. The NodeB, \(RX\), receives a signal that is the sum of the UE transmission \(TX\) and the interference transmission \(JX\). The resulting channel matrix, after filtering the AWGN, is therefore  
\[\left[ \begin{array}{cc}v_{1}&v_{2}\end{array} \right]+\left[ \begin{array}[] {cc}w_{1}&w_{2}\end{array} \right] \begin{bmatrix}e^{i\beta_{1}}&0\\ 0&e^{i\beta_{2}}\end{bmatrix} \]  
which we assume to be \(d\times 2\), where \(v_{2}^{*}v_{1}=0\) and \(w_{2}^{*}w_{1}=0\). Recall that for two vectors \(u,v\in\mathbb{C}\), \(u,v\
eq 0\), we have \(\frac{|u^{*}v|}{\|u\|\cdot\|v\|}\leq 1\). Therefore, we can write \(\frac{\|u^{*}v|}{\|u\|\cdot\|v\|}=cos(\varphi)\), for some \(\varphi\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). The columns of the resulting matrix will stay orthogonal if the condition in Equation (8) is met, i.e.,  
\[\left(v_{2}+e^{i\beta_{2}}w_{2}\right)^{*}\left(v_{1}+e^{i\beta_{1}}w_{1} \right)=0.\]  
Let \(\epsilon>0\) be the measure for orthogonality that the system can bear. We can then say that the columns of the resulting matrix are orthogonal up-to \(\epsilon\) error, if  
\[\frac{\left|\left(v_{2}+e^{i\beta_{2}}w_{2}\right)^{*}\left(v_{1}+e^{i\beta_ {1}}w_{1}\right)\right|^{2}}{\left\|v_{2}+e^{i\beta_{2}}w_{2}\right\|^{2}\cdot \left\|v_{1}+e^{i\beta_{1}}w_{1}\right\|^{2}}<\epsilon. \tag{14}\]Let us decompose \(e^{i\theta_{1}}w_{1}=u_{1}+z_{1}\) and \(e^{i\theta_{2}}w_{2}=u_{2}+z_{2}\), where \(u_{1},u_{2}\in Span(v_{1},v_{2})\) and \(z_{1},z_{2}\perp Span(v_{1},v_{2})\). Thus,  
\[\left(v_{2}+e^{i\theta_{2}}w_{2}\right)^{*}\left(v_{1}+e^{i\theta _{1}}w_{1}\right)\] \[=(v_{2}+u_{2}+z_{2})^{*}(v_{1}+u_{1}+z_{1})\] \[=(v_{2}+u_{2})^{*}(v_{1}+u_{1})+z_{2}^{*}z_{1},\]  
and  
\[ \begin{cases}\left\|v_{1}+e^{i\theta_{1}}w_{1}\right\|^{2}=\left\|v_{1}+u_{1} \right\|^{2}+\left\|z_{1}\right\|^{2}\\ \left\|v_{2}+e^{i\theta_{2}}w_{2}\right\|^{2}=\left\|v_{2}+u_{2}\right\|^{2}+ \left\|z_{2}\right\|^{2}.\end{cases}