## 2 System Model
Likewise, \(\mathbf{s}^{Q}\) denotes the imaginary elements in the modulated signal vector \(\mathbf{s}\), which in turn can be written as \([s_{1}^{Q},s_{2}^{Q},\ldots,s_{p}^{Q}]\). For \(I^{Q}\) and \(I^{I}\), respectively, assume that the \(\beta\)th legal AAPs are \(I_{\beta}^{I}\) and \(I_{\alpha}^{I}\), and may be stated as  
\[I_{a}^{I}=\{i_{a}^{I}(1),i_{a}^{I}(2),\ldots,i_{a}^{I}(p)\}, \tag{1}\]  
and  
\[I_{\beta}^{Q}=\{i_{\beta}^{Q}(1),i_{\beta}^{Q}(2),\ldots,i_{\beta}^{Q}(p)\}, \tag{2}\]  
Therefore, for the numbers \(\eta\) that lie between one and \(p\), \(i_{a}^{I}(\eta)\) and \(i_{\beta}^{Q}(\eta)\) are part of the collection of integers ranging from \(1\) to \(N_{t}\). The \(N_{t}\times 1\) signal vectors for the real and imaginary sections are as follows:  
\[\mathbf{x}^{I}=\{s_{1}^{I},0,s_{2}^{I},\ldots 0,s_{p}^{I}\}, \tag{3}\]  
and  
\[\mathbf{x}^{Q}=\{0,s_{1}^{Q},s_{2}^{Q},0,\ldots 0,s_{p}^{Q}\}, \tag{4}\]  
Thus, Keep in mind that AAPs \(I^{I}\) and \(I^{Q}\) decide the non-zero elements in \(\mathbf{x}^{I}\) and \(\mathbf{x}^{Q}\). The transmission is the last step in creating the signal vector, which has a dimension of \(N_{t}\times 1\). \[\mathbf{x}=\mathbf{x}^{I}+i\mathbf{x}^{Q} \tag{5}\]  
According to [14], the GQSM symbol, which comprises real, imaginary, along with values of zero for antenna indices, is represented by the symbol \(\mathbf{x}\) in (5). For \(N_{t}\times 1\), the transmit GQSM symbol is \(\mathbf{x}\); it also has a covariance matrix and a zero mean. \(\mathbf{R}_{\mathbf{x}}=[\mathbf{E}\mathbf{x}^{H}]=\sigma_{\mathbf{x}}^{2} \mathbf{I}\)  
Using a covariance matrix and a zero average \(\mathbf{R}_{n}=\mathbb{E}[\mathbf{n}\mathbf{n}^{H}]=\sigma_{n}^{2}\mathbf{I}\), we have the complex Gaussian noise vector \(\mathbf{n}\). The structure of the matrix represents the MIMO channel  
\[\mathbf{H}=[h_{j,k}\triangleq\alpha_{j,k}e^{i\phi_{jk}}]_{j,k=1}^{N_{t}N_{t}} \tag{6}\]  
where the potential mutually correlated channel coefficients are denoted by \(h_{j,k}\) and by \(i^{2}=-1\). The path gains \(\alpha_{j,k}\) of \(\kappa\triangleq n_{\tau}n_{R}\) are Nakagami distributed, as per the probability density function (PDF) provided by [30]. \[p_{\alpha_{j,k}}(\alpha)=2\sigma_{j,k}^{-2m}\frac{\alpha^{2m-1}}{\Gamma(m)} \exp\left(-\frac{\alpha^{2}}{\sigma_{j,k}^{2}}\right),\alpha\geq 0. \tag{7}\]  
The function \(\Gamma(. )\) represents the Gamma function [31]. The parameter m, with m being greater than or equal to \(1/2\), captures the severity of fading in the Nakagami model. The term \(\sigma_{j,k}^{2}\) is defined as the expectation of \(\alpha_{j,k}^{2}\) divided by m, in the same \(E[. ]\) represent the expectation operator. The phase \(\phi_{j,k}\) associated with the channel coefficient \(h_{j,k}\) follows a uniform distribution over the interval \([0,2\pi]\). The reason we chose the Nakagami fading model is that it reflects a wide range of multipath channels using the parameter m [30]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Index** & **AAP** & **Index** & **AAP** & **Index** & **AAP** & **Index** & **AAP** \\
**Bit** & **AAP** & **Bit** & **AAP** & **Bit** & **AAP** & **Bit** & **AAP** \\ \hline
00 & \((1,2)\) & \(10\) & \((1,4)\) & \(01\) & \((1,3)\) & \(11\) & \((2,3)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: GQSM mapping table with \(N_{t}=4\), \(m=4-QAM\) and \(p=2\). The PDF Equation (7) represents the one-sided Gaussian distribution at \(m=0.5\), but at \(m=1\), it represents the Rayleigh distribution. The Nakagami distribution provides a good approximation of the Rice distribution for \(m>1\) because the Nakagami fading parameter m and the Rician factor K are mapped one to one. \(\mathbf{h}=[h_{1},h_{2},h_{3},h_{4}]^{T}\) is the matrix where the partially static spatially correlated (associated) Nakagami m-fading channel coefficient of the \(j\)-th sending out (transmitting) antenna to the receiving antenna is denoted by \(h_{k}\), for \(j=1,\ldots,4\). The associated channel matrix \(\mathbf{H}\) is indicated by the matrix.