## 3 Conventional Data Detection Algorithm Based on Manifold Optimization
### Problem Formulation and Solving
Specifically, we compare the pilots obtained by solving the optimization problem with the real pilots to control when the iteration ends and thus maintain the performance of the estimation. When the two kinds of pilots are equal, the iteration is terminated. It should be pointed out that the pilot allocation in the proposed method is different from that in Section 3, where all pilots are placed at the first \(N_{t}\) OFDM symbols within one frame. In the proposed method, the pilots are discretely distributed within one frame. Let \(\mathcal{P}_{i}^{p}=\left\{\left(n_{i}^{p},k_{i}^{p}\right)\right\}\) denote the set of the pilot positions at the \(p\)th transmit antenna in the \(i\)th subframe and \(\left\{x_{n_{i}^{p},k_{i}^{p}}^{p}\right\}\) denote the corresponding transmitted pilot symbols, where \(n_{i}^{p}\in\{1+(i-1)B,\cdots,iB\}\), \(k_{i}^{p}\in\{1,\cdots,K\}\), and \(i=1,\cdots,N_{B}\). In the simulation, we set the position of the guide frequency as follows:

\[n_{i}^{p}=a,a+N,\cdots,a+\left(\lceil N_{f}/N\rceil-1\right)N, \tag{47}\]

\[k_{i}^{p}=p,p+M,\cdots,p+(\lceil K/M\rceil-1)M, \tag{48}\]

In practice, we actually use the values of \(a=1\), \(N=4\), and \(M=6\). Instead of using the complete OFDM symbols as the pilots when performing pilots, the pilots are inserted in OFDM symbols spaced at several fixed positions. The procedure of the proposed less-complex algorithm is summarized in Algorithm 3. In Algorithm 3, we estimate the signals from each subframe successively. Specifically, after solving the optimization problem to obtain the estimation of the transmit signals in the current subframe, we compare the symbols at the pilot positions with the true pilots. If the two kinds of pilots are equal, we continue to estimate the transmit signals of the next subframe. Otherwise, we take the estimated signal as a starting point and continue to estimate transmit signals of this subframe by solving the optimization problem. We repeat this process until the two kinds of pilots are equal. The flow diagram corresponding to Algorithm 3 is shown in Figure 1.

Next, we analyze the computational complexity of Algorithms 2 and 3. We use the number of complex multiplications to represent computational complexity. The biggest difference between Algorithms 2 and 3 lies in the number of iterations required in the Riemannian conjugate gradient algorithm, as well as the different dimensions of the matrices optimized in each iteration. The amount of computation of solving the optimization problem mainly comes from the calculation of the objective function \(F(\mathbf{\tilde{S}})\) and its gradient \(\text{Grad}\,F(\mathbf{\tilde{S}})\), where most of the computations involve matrix multiplications. Therefore, the difference in computational complexity between the two algorithms mainly lies in the computational complexity and number of calculations required for \(F(\mathbf{\tilde{S}})\) and \(\text{Grad}\,F(\mathbf{\tilde{S}})\). We analyze the complexity of \(F(\mathbf{\tilde{S}})\) and \(\text{Grad}\,F(\mathbf{\tilde{S}})\) in Algorithm 2 first. Some matrix multiplications appear repeatedly in different calculations, so they can be calculated only once. The required computational operations and their corresponding computational complexity are listed in Table 1. The most frequently occurring multiplicationis \(\mathbf{\tilde{H}}\mathbf{\tilde{S}}\), which has a complexity of \(\mathrm{O}\Big{(}4K^{3}{N_{r}}^{2}{N_{t}}{N_{f}}^{3}\Big{)}\). The multiplication \(\mathbf{\tilde{H}}\mathbf{\tilde{S}}\mathbf{\tilde{Y}}^{T}\) has a complexity of \(\mathrm{O}\Big{(}4K^{3}{N_{r}}^{3}{N_{f}}^{3}\Big{)}\). The multiplication \(\mathbf{\tilde{H}}\mathbf{\tilde{S}}\mathbf{\tilde{(}}\mathbf{\tilde{H}}\mathbf{\tilde{S}}\mathbf{ \tilde{)}}^{T}\) also has a complexity of \(\mathrm{O}\Big{(}4K^{3}{N_{r}}^{3}{N_{f}}^{3}\Big{)}\). Therefore, the complexity of calculating \(F(\mathbf{\tilde{S}})\) is \(\mathrm{O}\Big{(}4K^{3}{N_{r}}^{2}({N_{t}}+{N_{r}}){N_{f}}^{3}\Big{)}\).