## 3 Detection Techniques
### Detection Technique 1  
The received vector for detection is articulated in Equation (13), incorporating the multiplication of the channel matrix between the transmitting and receiving antennas, the signal vector, and the received noise.  
\[\mathbf{y}=\hat{\mathbf{H}}\mathbf{x}+\mathbf{n}, \tag{13}\]  
where, \(\mathbf{x}=\mathbf{x}^{I}+j\mathbf{x}^{Q}\).  
One way to factorize the relevant channel matrix is as follows.  
\[\hat{\mathbf{H}}=\mathbf{Q}_{1}\mathbf{R}_{1}=\mathbf{Q}_{1}\left[\begin{array}{cccc}a&0&b&0 \\ 0&a&0&b\\ 0&0&c&0\\ 0&0&0&c\end{array}\right] \tag{14}\]  
The upper diagonal matrix is represented by \(\mathbf{R}_{1}\), whereas the unitary matrix is shown by \(\mathbf{Q}_{1}\). On each side of (13), \(\mathbf{Q}_{1}^{H}\) gets multiplied to the left. The matrix that is produced is,  
\[\left[\begin{array}{c}\tilde{y}_{1}\\ \tilde{y}_{2}\\ \tilde{y}_{3}\\ \tilde{y}_{4}\end{array}\right]=\left[\begin{array}{cccc}a&0&b&0\\ 0&a&0&b\\ 0&0&c&0\\ 0&0&0&c\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{array}\right]+\left[\begin{array}{c}\tilde{n}_{1}\\ \tilde{n}_{2}\\ \tilde{n}_{3}\\ \tilde{n}_{4}\end{array}\right] \tag{15}\]  
Using (11), the \(\hat{\mathbf{H}}\mathbf{x}\) can be expressed as  
\[\left[\begin{array}{cccc}h_{4}&h_{3}&h_{2}&h_{1}\\ -h_{3}^{*}&h_{4}^{*}&-h_{1}^{*}&h_{2}^{*}\\ h_{2}&h_{1}&h_{4}&h_{3}\\ -h_{1}^{*}&h_{2}^{*}&-h_{3}^{*}&h_{4}^{*}\end{array}\right]\left[\begin{array} []{c}x_{4}\\ x_{3}\\ x_{2}\\ x_{1}\end{array}\right] \tag{16}\]  
Consequently, we obtain  
\[\mathbf{y}=\hat{\mathbf{H}}\mathbf{x}+\mathbf{n}=\hat{\mathbf{H}}^{\prime}\mathbf{x}^{\prime}+\mathbf{n} \tag{17}\]  
To find the \(\hat{\mathbf{H}}^{\prime}\) QR decomposition,  
\[\hat{\mathbf{H}}^{\prime}=\mathbf{Q}_{2}\mathbf{R}_{2}=\mathbf{Q}_{2}\left[\begin{array}{cccc }m&0&n&0\\ 0&m&0&n\\ 0&0&k&0\\ 0&0&0&k\end{array}\right] \tag{18}\]\(Q_{2}^{H}\) multiplied by the sides to the left and right of (17) produces  
\[\left[\begin{array}{c}\dot{y}_{1}^{\prime}\\ \dot{y}_{2}^{\prime}\\ \dot{y}_{3}^{\prime}\\ \dot{y}_{4}^{\prime}\end{array}\right]=\left[\begin{array}{cccc}m&0&n&0\\ 0&m&0&n\\ 0&0&k&0\\ 0&0&0&k\end{array}\right]\left[\begin{array}{c}x_{4}\\ x_{3}\\ x_{2}\\ x_{1}\end{array}\right]+\left[\begin{array}{c}\dot{n}_{1}^{\prime}\\ \dot{n}_{2}^{\prime}\\ \dot{n}_{3}^{\prime}\\ \dot{n}_{4}^{\prime}\end{array}\right], \tag{19}\]  
which can be stated as  
\[\vec{y}^{\prime}=\vec{R}_{2}\vec{x}^{\prime}+\vec{n}^{\prime} \tag{20}\]  
The 3 rd along with 4 th row of the (19) is combined with the 3 rd and 4 th of the (15) to create a new matrix, which brings about  
\[\left[\begin{array}{c}\ddot{y}_{4}^{\prime}\\ \dot{y}_{3}^{\prime}\\ \ddot{y}_{3}\\ \ddot{y}_{3}\\ \ddot{y}_{4}\end{array}\right]=\left[\begin{array}{cccc}k&0&0&0\\ 0&k&0&0\\ 0&0&c&0\\ 0&0&0&c\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{array}\right]+\left[\begin{array}{c}\ddot{n}_{1}^{\prime}\\ \dot{n}_{2}^{\prime}\\ \ddot{n}_{3}\\ \ddot{n}_{4}\end{array}\right] \tag{21}\]  
The four symbols are now separated as shown in (19) and (15). Consequently, parallel detection of \(x_{1},x_{2},x_{3}\), and \(x_{4}\) is possible.  
\[\dot{x_{1}}=\arg\min_{x_{1}\in\xi}\|\dot{y}_{4}^{\prime}-kx_{1}\|^{2}\quad\ \ \dot{x_{2}}=\arg\min_{x_{2}\in\xi}\|\dot{y}_{3}^{\prime}-kx_{2}\|^{2} \tag{22}\]  
\[\dot{x_{3}}=\arg\min_{x_{3}\in\xi}\|\dot{y}_{3}-cx_{3}\|^{2}\quad\ \ \dot{x_{4}}=\arg\min_{x_{4}\in\xi}\|\dot{y}_{4}-cx_{4}\|^{2} \tag{23}\]  
\(\dot{x_{1}},\dot{x_{2}},\dot{x_{3}}\) and \(\dot{x_{4}}\) are employed to identify the GQSM symbol.