# Study of 5G-NR-MIMO Links in the Presence of an Interferer
## 3 Alamouti MIMO Link in the Presence of Different Interferer Techniques
### Smart Jammer  
When we describe the smart jammer principle, it is critical to know the RAN behavior and the protocols that manage the PHY layer of the target wireless communication system's air interface. A smart jammer has two main guiding principles. The first one is the ability to estimate the channel matrix or the CSI between the UE to the base-station (e/gNodeB), and between the jammer to this base-station (NodeB), from the jammer's perspective [21]. An example is a smart jammer technique called a pilot-nulling attack. The adaptive process is called Battle Damage Assessment (BDA) [43]. Another example is a jammer that joins a network to legitimize the UE and manipulate the RF PHY layer. This is very destructive and effective against the communication system. The second principle is communication deterioration by converting the target system's advantage to the most significant disadvantage [27]. An excellent example of this scenario is the exploitation of the channel's multiplicity of paths that characterizes Rayleigh fading. This exploitation is manifested in the fact that, with advanced MIMO techniques, we cause in the receiver processing a situation of linear independence between every path [27]. If the smart jammer hits this principle, many interference signals will be involved in the de-modulator and the combiner at the receive side. It leads to a destructive chain reaction that begins with a decrease in the modulation order that extends the transmission time and causes a decrease in BER performance.  
In this subsection, we describe the communication model of combining a smart jammer using the Alamouti-STC MIMO model, which is a \(2\times 2\) MIMO array. Based on the assumption that the smart jammer is estimating the channel matrix, \(H_{TR}\), between the UE to the base-station (NodeB), we can write Equation (11) as  
\[\bar{R}=\begin{bmatrix}r_{1}(1)\\ r_{1}^{*}(2)\\ r_{2}(1)\\ r_{2}^{*}(2)\end{bmatrix}=\sqrt{P}\begin{bmatrix}h_{11}&h_{12}\\ h_{12}^{*}&-h_{11}^{*}\\ h_{21}^{*}&h_{22}\\ h_{22}^{*}&-h_{21}^{*}\end{bmatrix}\begin{bmatrix}s_{1}\\ s_{2}\end{bmatrix}+\sqrt{\frac{P}{SIR}}\begin{bmatrix}h_{11}&h_{12}\\ h_{12}^{*}&-h_{11}^{*}\\ h_{21}^{*}&h_{22}^{*}\\ h_{22}^{*}&-h_{21}^{*}\end{bmatrix}\begin{bmatrix}j_{1}\\ j_{2}\end{bmatrix}+\sqrt{\frac{P}{SNR}}\begin{bmatrix}n_{1}(1)\\ n_{1}^{*}(2)\\ n_{2}^{*}(1)\\ n_{2}^{*}(2)\end{bmatrix} \tag{14}\]  
After normalizing the powers, the receive equations are  
\[\begin{split} r_{1}(1)&=h_{11}(s_{1}+j_{2})+h_{12}(s_{2}+j_{1} )+n_{1}(1)\\ r_{1}^{*}(2)&=h_{12}^{*}(s_{1}-j_{2})-h_{11}^{*}(s_{2}-j_{1})+n_{1}^{*}(2) \\ r_{2}(1)&=h_{21}(s_{1}+j_{2})+h_{22}(s_{2}+j_{1})+n_{2}(1)\\ r_{2}^{*}(2)&=h_{22}^{*}(s_{1}-j_{2})-h_{21}^{*}(s_{2}-j_{1})+n_{2}^{*}(2) \end{split} \tag{15}\]  
If we are substituting (15) into the combiner rule at the receiver (2), it yields  
\[\tilde{s}_{1} = \left(\sum_{i=1}^{2}\sum_{j=1}^{2}\left|h_{ij}\right|^{2}\right) s_{1}+h_{11}^{*}n_{1}(1)+h_{12}n_{1}^{*}(2)+h_{21}^{*}n_{2}(1)+h_{22}n_{2}^{*}(2)+\] \[+j_{2}\Big{(}\left|h_{11}\right|^{2}-\left|h_{12}\right|^{2}+ \left|h_{21}\right|^{2}-\left|h_{22}\right|^{2}\Big{)}\]  
and  
\[\begin{split}\tilde{s}_{2}&=& \left(\sum_{i=1}^{2}\sum_{j=1}^{2}\left|h_{ij}\right|^{2}\right) s_{2}+h_{12}^{*}n_{1}(1)-h_{11}n_{1}^{*}(2)+h_{22}^{*}n_{2}(1)-h_{21}n_{2}^{*}(2)+ \\ &+j_{1}\Big{(}-\left|h_{11}\right|^{2}+\left|h_{12}\right|^{2}- \left|h_{21}\right|^{2}+\left|h_{22}\right|^{2}\Big{)}\end{split} \tag{16}\]  
The result obtained is impressive because the interference, multiplied by the norms, was obtained in addition to the symbol, i.e., the interfering symbol was added. In Section 4, we will present simulation results for representative cases of this phenomenon.