## 4 Bit-Error-Rate Analysis  
Using the OGSMP-MIMO system with the number of transmit antennas \(N_{t}=6\) and the receiving antennas \(N_{r}=6\) as an example, \(L\)-PAM was adopted as the modulation method, and the BER performance of the OGSMP-MIMO system was analyzed and deduced, using segmental bound theory. Assuming that the total error bits in the OGSMP-MIMO system are \(m_{c}\), and the total data sent is \(10^{6}\), then the system BER can be divided into index-position and constellation-symbol errors. The error bits caused by the index-position and constellation-symbol errors are denoted as \(m_{e0}\) and \(m_{e1}\), respectively. From Equation (4), the original bit stream signal can be determined; then, the received signal is restored as  
\[\hat{x}_{u}=\left[0,\cdots,0,\hat{A}_{u},0,\cdots,0\right]^{T}\ \ \hat{x}_{u}\in C_{10^{6}\times 1} \tag{16}\]  
Considering \(P(x_{u}\rightarrow\hat{x}_{u}|H)\) to represent the conditional probability that the transmitted signal is \(x_{u}\) and the receiving end recovers to \(\hat{x}_{u}\) when the channel information \(H\) is known, then using the union bound theory, the upper bound can be expressed as  
\[m_{c}\leq\sum_{u}\sum_{\hat{u}}P(x_{u}\rightarrow\hat{x}_{u}|H)\frac{1}{10^{6 }}\hat{m}_{e}(u,\hat{u}) \tag{17}\]  
where \(\hat{m}_{e}(u,\hat{u})\) represents the Hamming distance between \(x_{u}\) and \(\hat{x}_{u}\), and the expression form of the signal at the receiving end, according to \(y_{u}=H_{u}x_{u}+n_{u}\), is  
\[\mathbf{y}_{u}=\left[n_{1},\cdots,H_{u}A_{u}+n_{u},\cdots,n_{10^{6}}\right] \tag{18}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Input Bit**} & **Antenna Index** & **PAM Mapping** & **Output Signal** \\  & **Symbol** & **Modulation Symbol** & **after Mapping** \\ \hline
00000 & \([1\,1\,0\,0\,0\,0\,0]^{T}\) & \(\frac{2}{3},\frac{2}{3}\) & \(\left[\frac{\sqrt{2}}{3},\frac{\sqrt{2}}{3},0,0,0,0\right]^{T}\) \\
00001 & \([1\,1\,0\,0\,0\,0\,0]^{T}\) & \(\frac{2}  
When the receiving end returns to \(\hat{\mathbf{x}}_{u}\), i.e., the antenna position index is incorrect, the conditions should be met at this time. \[\left\|y_{u}-H_{u}x_{u}\right\|_{F}^{2}>\left\|y_{u}-\hat{H}_{u}\hat{x}_{u} \right\|_{F}^{2} \tag{19}\]  
Equation (20) can be simplified to  
\[2\text{Re}\Big{\{}\left(\hat{H}_{u}\right)^{H}\hat{n}_{u}\Big{\}}-2\text{Re} \Big{\{}\left(H_{u}\right)^{H}n_{u}\Big{\}}>\left|H_{u}A_{u}\right|^{2}+\left| \hat{H}_{u}\hat{A}_{u}\right|^{2} \tag{20}\]  
where \(\text{n}_{u}\) represents the complex Gaussian noise of the \(u\)-th antenna, and defines \(N=2\text{Re}\Big{\{}\left(\hat{H}_{u}\right)^{H}\hat{n}_{u}\Big{\}}-2\text{Re }\Big{\{}\left(H_{u}\right)^{H}n_{u}\Big{\}}\); accordingly, it can be deduced that it is a zero-mean Gaussian random variable:  
\[N\sim N\Big{(}0,2\left|H_{u}A_{u}\right|^{2}\cdot N_{0}+2\left|\hat{H}_{u}\hat {A}_{u}\right|^{2}\cdot N_{0}\Big{)} \tag{21}\]  
Hence,  
\[P(x_{u}\rightarrow\hat{x}_{u}\left|H\right)=Q\Bigg{(}\sqrt{\frac{\left|H_{u}A _{u}\right|^{2}+\left|\hat{H}_{u}\hat{A}_{u}\right|^{2}}{2N_{0}}}\Bigg{)} \tag{22}\]  
where \(Q(x)=\int\limits_{x}^{\infty}\exp\left(-t^{2}/2\right)/\sqrt{2\pi}dt\) and \(P(x_{u}\rightarrow\hat{x}_{u}\left|H\right)\) denotes the conditional probability. In the actual simulation, the channel information generated in the simulation is substituted into Equation (23), and the final pairwise error probability \(P(x_{u}\rightarrow\hat{x}_{u}\left|H\right)\) is obtained by averaging after multiple simulations. We denote \(e(u,\hat{u})\) as the number of erroneous index bits when ascertaining the index position of the active antenna as \(\hat{u}\), i.e., the Hamming distance between the two. Hence  
\[m_{e0}=\frac{1}{10^{6}}\sum\limits_{u=1}^{10^{6}}\sum\limits_{\hat{u}\
eq u=1 }^{10^{6}}P(x_{u}\rightarrow\hat{x}_{u})e(u,\hat{u}) \tag{23}\]  
The constellation symbol error comprises two parts: \(m_{e10}\) when the index position is incorrect, and \(m_{e11}\), when the index position is correct.