## 3 Conventional Data Detection Algorithm Based on Manifold Optimization
### Problem Formulation and Solving
So the complexity of the improved Algorithm 3 is 0.73% of Algorithm 2. Figure 1: Flow diagram of Algorithm 3. ## 5 Simulation Results and Discussions

We evaluate the performance of the proposed Algorithms 2 and 3 through simulations and discuss implications of symbol number \(B\) in one subframe on the performance of Algorithm 3. In addition, we compare the BER of the proposed algorithms with that of the traditional ZF detection. In the simulation, we use the MIMO-OFDM system parameters set as shown in Table 3. We conduct simulations in a MIMO-OFDM system with \(N_{t}=4\) transmit antennas, \(N_{r}=4\) receive antennas, \(K=64\) subcarriers, and a data frame length of \(N_{f}=14\). The channel model we employ is a time-frequency variation TDL-B channel with 12 taps [17], a Doppler shift of 400 Hz, and a sampling rate of 30.72 MHz. Under such channel conditions, Algorithm 1 cannot be applied. Figure 2 shows the BER of proposed algorithms with different number of symbols \(B\) of 1, 2, and 7 under QPSK modulation. The performance of the conventional ZF detection is also shown in the figure for comparison. From Figure 2, we can see that the BER values decrease for all algorithms as SNR increases. The performance of Algorithm 3 is significantly better than direct ZF detection. When BER is \(10^{-2}\), Algorithm 3 with \(B=1\) has an approximately 5 dB SNR gain compared to the direct ZF detection algorithm. The simulation also shows the performances of Algorithm 3 with \(B=1\) and \(B=2\) are relatively similar and Algorithm 3 with \(B=7\) performs better under high-SNR conditions. However, according to the discussion in Section 4.2, there is a significant difference in the complexity of Algorithm 3 with different \(B\) values. The smaller the number of symbols \(B\), the lower \begin{table}
\begin{tabular}{l l} \hline \hline
**Computational Operations** & **Computational Complexity** \\ \hline \(HS\) & \(\mathrm{O}\left(4K^{3}{N_{r}}^{2}N_{t}N_{f}{}^{3}\right)\) \\ \(\tilde{H}\tilde{S}\tilde{Y}^{T}\) & \(\mathrm{O}\left(4K^{3}{N_{r}}^{3}N_{f}{}^{3}\right)\) \\ \(\tilde{H}\tilde{S}(\tilde{H}\tilde{S})^{T}\) & \(\mathrm{O}\left(4K^{3}{N_{r}}^{3}N_{f}{}^{3}\right)\) \\ \(\tilde{H}^{T}\tilde{Y}\) & \(\mathrm{O}\left(4K^{3}{N_{r}}^{2}N_{t}N_{f}{}^{3}\right)\) \\ \(\tilde{H}^{T}\tilde{H}\tilde{S}\) & \(\mathrm{O}\left(4K^{3}{N_{r}}^{2}N_{t}N_{f}{}^{3}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Algorithm 2 computational complexity analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Subframe Length \(B\)** & **Computational Complexity** \\ \hline Algorithm 2 & \(\sim\) & \(\mathrm{O}\left(76832K^{3}{N_{r}}^{2}(2N_{t}+N_{r})\right)\) \\ \hline \multirow{3}{*}{Algorithm 3} & 1 & \(\mathrm{O}\left(560K^{3}{N_{r}}^{2}(2N_{t}+N_{r})\right)\) \\ \cline{2-3}  & 2 & \(\mathrm{O}\left(2240K^{3}{N_{r}}^{2}(2N_{t}+N_{r})\right)\) \\ \cline{2-3}  & 7 & \(\mathrm{O}\left(27440K^{3}{N_{r}}^{2}(2N_{t}+N_{r})\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison table of the computational complexity of algorithms with \(N_{f}=14\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Simulation Parameter Table** & \\ \hline Transmit Antennas \(N_{t}\) & 4 \\ \hline Receive Antennas \(N_{r}\) & 4 \\ \hline Subcarriers \(K\) & 64 \\ \hline Data Frame Length \(N_{f}\) & 14 \\ \hline Doppler Shift/Hz & 400 \\ \hline Sampling Rate/MHz & 30.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. the complexity. The complexity of Algorithm 3 is approximately \(\text{O}(2.8\times 10^{10})\) when \(B=1\), \(\text{O}(11.3\times 10^{10})\) when \(B=2\), and \(\text{O}(138.1\times 10^{10})\) when \(B=7\). By comparing the performances of Algorithms 2 and 3, it can be noted that the BER performance of Algorithm 3 with the number of symbols of 1 and Algorithm 2 is similar. However, the complexity of Algorithm 3 with \(B=1\) is 0.73% of Algorithm 2.