## 2 System Model
\(T\) received vector samples can be combined into a single matrix as  
\[\mathbf{Y}=[\mathbf{y}[k]\ldots\mathbf{y}[k+T-1]] =\sqrt{\frac{P_{T}}{N_{T}}} \begin{bmatrix}\mathbf{h}_{1,1}&\ldots& \mathbf{h}_{1,N_{T}}\\ \vdots&\vdots&\vdots\\ \mathbf{h}_{N_{R},1}&\ldots&\mathbf{h}_{N_{R},N_{T}}\end{bmatrix}\begin{bmatrix} \mathbf{X}_{1}\\ \vdots\\ \mathbf{X}_{N_{T}}\end{bmatrix} +\mathbf{V} \tag{8}\] \[=\sqrt{\frac{P_{T}}{N_{T}}}\mathbf{H}\mathbf{X}+\mathbf{V},\]  
where \(\mathbf{X}_{j}\) and \(\mathbf{V}\) are written as  
\[\mathbf{X}_{j}= \begin{bmatrix}x_{j}[k-L+1]&x_{j}[k-L+2]&\ldots&x_{j}[k-L+T]\\ \vdots&\vdots&\ldots&\vdots\\ x_{j}[k-1]&x_{j}[k]&\ldots&x_{j}[k+T-2]\\ x_{j}[k]&x_{j}[k+1]&\ldots&x_{j}[k+T-1]\end{bmatrix} \text{ and } \tag{9}\]  
\[\mathbf{V}= \begin{bmatrix}v_{1}[k]&v_{1}[k+1]&\ldots&v_{1}[k+T-1]\\ v_{2}[k]&v_{2}[k+1]&\ldots&v_{2}[k+T-1]\\ \vdots&\vdots&\ldots&\vdots\\ v_{N_{R}}[k]&v_{N_{R}}[k+1]&\ldots&v_{N_{R}}[k+T-1]\end{bmatrix} . \tag{10}\]  
After receiving \(\mathbf{Y}\), the Minimum Mean Squared Error (MMSE) equalizer is used to mitigate the negative effects caused by the channel, such as fading. The MMSE equalization matrix, \(\mathbf{W}_{MMSE}\)[49], is calculated as  
\[\mathbf{W}_{MMSE} =\sqrt{\frac{N_{T}}{P_{T}}}\bigg{(}\mathbf{H}^{H}\mathbf{H}+ \frac{P_{V}N_{T}}{P_{T}}\mathbf{I}_{N_{T}}\bigg{)}^{-1}\mathbf{H}^{H} \tag{11}\] \[=\sqrt{\frac{N_{T}}{P_{T}}}\mathbf{B}\mathbf{H}^{H},\]where \(P_{V}=P_{J}+P_{N}\) is the sum of jamming (\(P_{J}\)) and noise (\(P_{N}\)) powers and  
\[\mathbf{B}=\bigg{(}\mathbf{H}^{H}\mathbf{H}+\frac{P_{V}N_{T}}{P_{T}}\mathbf{I}_{ N_{T}}\bigg{)}^{-1}. \tag{12}\]  
To estimate the transmitted IQ symbols, the equalizer is applied as follows:  
\[ \begin{split}\mathbf{\hat{X}}=\mathbf{W}_{MMSE}\mathbf{Y}& =\sqrt{\frac{N_{T}}{P_{T}}}\mathbf{B}\mathbf{H}^{H}\mathbf{Y}\\ &=\mathbf{B}\mathbf{H}^{H}\mathbf{H}\mathbf{X}+\sqrt{\frac{N_{T} }{P_{T}}}\mathbf{B}\mathbf{H}^{H}\mathbf{V}\\ &=\mathbf{B}\mathbf{C}\mathbf{X}+\sqrt{\frac{N_{T}}{P_{T}}} \mathbf{B}\mathbf{H}^{H}\mathbf{V},\end{split}  \tag{13}\]  
where \(\mathbf{C}=\mathbf{H}^{H}\mathbf{H}\). At this point, the proposed error vector magnitude (EVM) metric for jamming detection is calculated using the reference and estimated IQ symbols as follows:  
\[EVM_{n}=\sqrt{\frac{e_{n}^{2}}{\frac{1}{N}\sum_{n=1}^{N}(i_{n}^{2}+q_{n}^{2})}}, \tag{14}\]  
where  
* \(n\) denotes the index of the IQ symbol,
* \(N\) is the total number of symbols used for calculation,
* \(e_{n}^{2}=(i_{n}-\hat{i_{n}})^{2}+(q_{n}-\hat{q_{n}})^{2}\) is the power of the error caused by the jamming and noise,
* \(i_{n}\) and \(q_{n}\) are the reference in-phase and quadrature values of the \(n^{th}\) symbol (\(x_{n}=i_{n}+jq_{n}\)),
* \(\hat{i_{n}}\) and \(\hat{q_{n}}\) are the estimated in-phase and quadrature values of the \(n^{th}\) symbol (\(\hat{x_{n}}=\hat{i_{n}}+j\hat{q_{n}}\)),
* \(\frac{1}{N}\sum_{n=1}^{N}(i_{n}^{2}+q_{n}^{2})\) represents the average power of the reference symbols. As shown in Equation (14), each of the \(N\) symbols is used once for vectoral difference calculation. Therefore, the computational complexity of the EVM is in terms of the first power of \(N\), that is, \(O(N)\). Consequently, the computational complexity of our jamming detection method using the EVM metric has a low value of \(O(N)\). For EVM calculation, both reference and estimated symbols are required. The natural flow of next-generation communication systems, such as LTE and 5G, includes the transmission of reference symbols. In this manner, without any pre-training and without changing the system architecture, we calculate the EVM metric and detect the presence of a jamming signal by checking whether the EVM exceeds a single threshold level. This makes the proposed method very advantageous in terms of integrability into real-world scenarios. In addition, Equation (14) indicates that EVM is proportional to the square root of the jamming plus noise-to-signal ratio (\(JNSR_{\ell}\)).