## 3 Conventional Data Detection Algorithm Based on Manifold Optimization  
In this section, the conventional data detection algorithm for MIMO-OFDM systems based on manifold optimization is presented. Here, it is assumed that the channel is constant in a frame, i.e., \(\mathbf{H}_{1,k}=\cdots=\mathbf{H}_{N_{f},k}=\mathbf{H}_{k}\). By collecting all the data at the \(k\)th subcarriers in one frame, we can obtain  
\[\mathbf{Y}_{k}=\mathbf{H}_{k}\mathbf{X}_{k}+\mathbf{W}_{k}, \tag{3}\]  
where \(\mathbf{X}_{k}= \begin{bmatrix}\mathbf{x}_{1,k}&\mathbf{x}_{2,k}&\cdots&\mathbf{x}_{N_{f},k} \end{bmatrix} \in\mathbb{C}^{N_{t}\times N_{f}}\), \(\mathbf{Y}_{k}= \begin{bmatrix}\mathbf{y}_{1,k}&\mathbf{y}_{2,k}&\cdots&\mathbf{y}_{N_{f},k} \end{bmatrix} \in\mathbb{C}^{N_{t}\times N_{f}}\), and \(\mathbf{W}_{k}= \begin{bmatrix}\mathbf{w}_{1,k}&\mathbf{w}_{2,k}&\cdots&\mathbf{w}_{N_{f},k} \end{bmatrix} \in\mathbb{C}^{N_{t}\times N_{f}}\). Given \(\mathbf{Y}_{k}\), the joint estimation of \(\mathbf{X}_{k}\) and \(\mathbf{H}_{k}\) can be formulated as the following optimization problem [12]:  
\[\min_{\mathbf{H}_{k},\mathbf{X}_{k}}J(H_{k},\mathbf{X}_{k})=\left\|\mathbf{Y}_{k}-\mathbf{H}_{k} \mathbf{X}_{k}\right\|_{F}^{2}, \tag{4}\]  
where \(\left\|\cdot\right\|_{F}\) denotes the Frobenius norm. Since the manifold is defined on the real number domain, the above optimization problem needs to be transformed into a real-valued representation, which is given by [12]  
\[\min_{\mathbf{H}_{k},\mathbf{X}_{k}}J(\mathbf{\tilde{H}}_{k},\mathbf{\tilde{X}}_{k})=\left\| \mathbf{\tilde{Y}}_{k}-\mathbf{\tilde{H}}_{k}\mathbf{\tilde{X}}_{k}\right\|_{F}^{2}, \tag{5}\]  
where  
\[\mathbf{\tilde{H}}_{k}= \begin{bmatrix}\text{Re}(\mathbf{H}_{k})&-\text{Im}(\mathbf{H}_{k} )\\ \text{Im}(\mathbf{H}_{k})&\text{Re}(\mathbf{H}_{k})\end{bmatrix} \in\mathbb{R}^{2N_{t} \times 2N_{t}}, \tag{6}\]\[\tilde{\mathbf{\Upsilon}}_{k}= \begin{bmatrix}\text{Re}(\mathbf{Y}_{k})\\ \text{Im}(\mathbf{Y}_{k})\end{bmatrix} \in\mathbb{R}^{2N_{t}\times N_{f}}, \tag{7}\]  
\[\tilde{\mathbf{X}}_{k}= \begin{bmatrix}\text{Re}(\mathbf{X}_{k})\\ \text{Im}(\mathbf{X}_{k})\end{bmatrix} \in\mathbb{R}^{2N_{t}\times N_{f}}, \tag{8}\]  
\(\text{Re}(\cdot)\) denotes the real part and \(\text{Im}(\cdot)\) denotes the imaginary part. We can obtain the optimal \(\tilde{\mathbf{X}}_{k}\) and \(\tilde{\mathbf{H}}_{k}\) by iterative solution of the optimization problem (5). Given \(\tilde{\mathbf{X}}_{k}\), since \(\tilde{\mathbf{Y}}_{k}=\tilde{\mathbf{H}}_{k}\tilde{\mathbf{X}}_{k}+\tilde{\mathbf{W}}_{k}\), where \(\tilde{\mathbf{W}}_{k}= \begin{bmatrix}\text{Re}(\mathbf{W}_{k})\\ \text{Im}(\mathbf{W}_{k})\end{bmatrix} \in\mathbb{R}^{2N_{t}\times N_{f}}\), the least squares estimation of \(\tilde{\mathbf{H}}_{k}\) can be computed as  
\[\tilde{\mathbf{H}}_{k}=\tilde{\mathbf{Y}}_{k}\mathbf{X}_{k}^{T}(\tilde{\mathbf{X}}_{k}\mathbf{X}_{ k}^{T})^{-1}. \tag{9}\]  
Substituting (9) into (5), we can obtain the following simplified objective function:  
\[ \begin{split} J(\tilde{\mathbf{X}}_{k})&=\left\|\tilde{ \mathbf{Y}}_{k}-\tilde{\mathbf{Y}}_{k}\mathbf{X}_{k}^{T}\Big{(}\tilde{\mathbf{X}}_{k}\mathbf{X}_{ k}^{T}\Big{)}^{-1}\tilde{\mathbf{X}}_{k}\right\|_{F}^{2}\\ &=\text{trace}\Bigg{\{}\bigg{[}\tilde{\mathbf{Y}}_{k}-\tilde{\mathbf{Y}}_{ k}\tilde{\mathbf{X}}_{k}^{T}\Big{(}\tilde{\mathbf{X}}_{k}\tilde{\mathbf{X}}_{k}^{T}\Big{)}^{-1} \tilde{\mathbf{X}}_{k}\bigg{]}\bigg{[}\tilde{\mathbf{Y}}_{k}-\tilde{\mathbf{Y}}_{k}\tilde{ \mathbf{X}}_{k}^{T}\Big{(}\tilde{\mathbf{X}}_{k}\tilde{\mathbf{X}}_{k}^{T}\Big{)}^{-1} \tilde{\mathbf{X}}_{k}\bigg{]}^{T}\Bigg{\}}\\ &=\text{trace}\Big{\{}\tilde{\mathbf{Y}}_{k}\tilde{\mathbf{Y}}_{k}^{T} \Big{\}}-\text{trace}\Bigg{\{}\tilde{\mathbf{Y}}_{k}\tilde{\mathbf{X}}_{k}^{T}\Big{(} \tilde{\mathbf{X}}_{k}\mathbf{X}_{k}^{T}\Big{)}^{-1}\tilde{\mathbf{X}}_{k}\tilde{\mathbf{Y}}_ {k}^{T}\Bigg{\}},\end{split}  \tag{10}\]  
where \(\text{trace}\{\cdot\}\) denotes the trace.