## 3 Conventional Data Detection Algorithm Based on Manifold Optimization
### Problem Formulation and Solving  
By collecting the received signals at all subcarriers of the \(n\)th symbol, we obtain  
\[\mathbf{Y}_{n}=\mathbf{H}_{n}\mathbf{X}_{n}+\mathbf{W}_{n}, \tag{22}\]  
where \(\mathbf{Y}_{n}=\text{diag}\{ \begin{bmatrix}\mathbf{y}_{n,1}&\mathbf{y}_{n,2}&\cdots&\mathbf{y} _{n,K}\end{bmatrix} \}\in\mathbb{C}^{\text{KN}_{n}\times K},\mathbf{X}_{n}=\text{ diag}\{ \begin{bmatrix}\mathbf{x}_{n,1}&\mathbf{x}_{n,2}&\cdots&\mathbf{x}_{n,K}\end{bmatrix} \}\), \(\in\mathbb{C}^{\text{KN}_{n}\times K}\), \(\mathbf{W}_{n}=\text{diag}\{ \begin{bmatrix}\mathbf{w}_{n,1}&\mathbf{w}_{n,2}&\cdots&\mathbf{w}_{ n,K}\end{bmatrix} \}\in\mathbb{C}^{\text{KN}_{n}\times K}\),  
\[\mathbf{H}_{n}=\text{diag}\{ \begin{bmatrix}\mathbf{H}_{n,1}&\mathbf{H}_{n,2}&\cdots&\mathbf{H} _{n,K}\end{bmatrix} \}\in\mathbb{C}^{\text{KN}_{n}\times\text{KN}_{n}}, \tag{23}\]  
\[\text{diag}\{ \begin{bmatrix}\mathbf{A}_{1}&\mathbf{A}_{2}&\cdots&\mathbf{A}_{K}\end{bmatrix}  \}\triangleq \begin{bmatrix}\mathbf{A}_{1}&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{0}&\mathbf{A}_{2}&\cdots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{0}&\mathbf{0}&\cdots&\mathbf{A}_{K}\end{bmatrix} , \tag{24}\]  
and \(\mathbf{A}_{k}\) denotes a matrix or vector. Considering all \(N_{f}\) OFDM symbols in one frame, we formulate the objective function as  
\[J(\mathbf{X}_{1},\cdots,\mathbf{X}_{N_{f}})=\sum_{n=1}^{N_{f}}\left\|\dot{\mathbf{Y}}_{n}- \dot{\mathbf{H}}_{n}\dot{\mathbf{X}}_{n}\right\|_{F}^{2}. \tag{25}\]  
Similarly to Section 3, we transform the optimization problem into a real-valued representation, which is given by  
\[J(\dot{\mathbf{X}}_{1},\cdots,\dot{\mathbf{X}}_{N_{f}})=\sum_{n=1}^{N_{f}}\left\|\dot {\mathbf{Y}}_{n}-\dot{\mathbf{H}}_{n}\dot{\mathbf{X}}_{n}\right\|_{F}^{2}. \tag{26}\]  
where  
\[\dot{\mathbf{X}}_{n}=\text{diag}\{ \begin{bmatrix}\mathbf{\bar{x}}_{n,1}&\mathbf{\bar{x}}_ {n,2}&\cdots&\mathbf{\bar{x}}_{n,K}\end{bmatrix} \}\in\mathbb{R}^{2\text{KN}_{l} \times K}, \tag{27}\]  
\[\dot{\mathbf{Y}}_{n}=\text{diag}\{ \begin{bmatrix}\mathbf{\bar{y}}_{n,1}&\mathbf{\bar{y}}_ {n,2}&\cdots&\mathbf{\bar{y}}_{n,K}\end{bmatrix} \}\in\mathbb{R}^{2\text{KN}_{l} \times K}, \tag{28}\]  
\[\dot{\mathbf{H}}_{n}=\text{diag}\{ \begin{bmatrix}\mathbf{H}_{n,1}&\mathbf{H}_{n,2}&\cdots &\mathbf{\bar{H}}_{n,K}\end{bmatrix} \}\in\mathbb{R}^{2\text{KN}_{l}\times 2\text{KN}_{l}}, \tag{29}\]  
\[\bar{\mathbf{x}}_{n,k}= \begin{bmatrix}\text{Re}(\mathbf{x}_{n,k})\\ \text{Im}(\mathbf{x}_{n,k})\end{bmatrix} \in\mathbb{R}^{2\text{N}_{l}\times 1}.\]According to (27)-(29), (26) is further expanded as  
\[ \begin{split} J(\tilde{\mathbf{x}}_{1,1},\cdots,\tilde{\mathbf{x}}_{N_{f},K} )&=\sum_{n=1}^{N_{f}}\sum_{k=1}^{K}\text{trace}\Big{\{}(\tilde{\mathbf{ y}}_{n,k}-\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k})(\tilde{\mathbf{y}}_{n,k}-\tilde{\mathbf{H}}_{n,k} \tilde{\mathbf{x}}_{n,k})^{T}\Big{\}}\\ &=\sum_{n=1}^{N_{f}}\sum_{k=1}^{K}\text{trace}\{\tilde{\mathbf{y}}_{n, k}\tilde{\mathbf{y}}_{n,k}^{T}-\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k}\tilde{\mathbf{y}}_{n,k}^ {T}-\tilde{\mathbf{y}}_{n,k}\tilde{\mathbf{x}}_{n,k}^{T}\tilde{\mathbf{H}}_{n,k}^{T}\\ &+\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k}\tilde{\mathbf{x}}_{n,k}^{T }\tilde{\mathbf{H}}_{n,k}^{T}\}.\end{split}