# Singular Value Decomposition Channel Estimation in STBC MIMO-OFDM System
## Appendix A  
Defining \(\hat{H}_{\text{LS}}=\mathbf{X}^{-1}\mathbf{Y}=\widetilde{\mathbf{H}}\), we denote the MMSE channel estimation by:  
\[\hat{\mathbf{H}}_{\text{MMSE}}=\mathbf{A}\widetilde{\mathbf{H}}, \tag{10}\]  
where \(\mathbf{A}\) is a weighting matrix. The MSE of MMSE channel estimation can be represented as:  
\[\text{E}(|\mathbf{e}|^{2})=\text{E}(|\mathbf{H}-\hat{\mathbf{H}}_{\text{MMSE}}|^{2}), \tag{11}\]  
where \(\mathbf{e}\) is the estimated error between the true CFR and estimated CFR based on the MMSE method. The MMSE channel estimation method has better performance than the LS and threshold methods since it is optimum in minimizing the MSE of the channel estimates in the presence of AWGN. The estimated error \(\mathbf{e}\) and \(\widetilde{\mathbf{H}}\) are orthogonal, which can be represented as:  
\[\begin{array}{ll}\text{E}(\mathbf{e}\widetilde{\mathbf{H}}^{\text{H}})&=\text{E}[( \mathbf{H}-\mathbf{\hat{H}})\widetilde{\mathbf{H}}^{\text{H}}]=\text{E}[(\mathbf{H}-\mathbf{A} \ \widetilde{\mathbf{H}})\widetilde{\mathbf{H}}^{\text{H}}]\\ &=\text{E}(\mathbf{H}\widetilde{\mathbf{H}}^{\text{H}})-\text{A}\text{E}(\widetilde{ \mathbf{H}}\widetilde{\mathbf{H}}^{\text{H}})=\mathbf{R}_{\widetilde{\mathbf{H}}\widetilde{ \mathbf{H}}}-\mathbf{A}\mathbf{R}_{\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}}\\ &=0\end{array} \tag{12}\]  
According to Equation (12), the weighting matrix \(\mathbf{A}\) can be represented as:  
\[\mathbf{A}=\mathbf{R}_{\widetilde{\mathbf{H}}}\widetilde{\mathbf{H}}^{\text{R}-1}, \tag{13}\]  
where \(\mathbf{R}_{\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}}^{-1}\) is the autocorrelation matrix of \(\widetilde{\mathbf{H}}\), which can be represented as:  
\[\begin{array}{ll}\mathbf{R}_{\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}}&=\text{E}( \widetilde{\mathbf{H}}\widetilde{\mathbf{H}}^{\text{H}})=\text{E}[\mathbf{X}^{-1}\mathbf{Y}( \mathbf{X}^{-1}\mathbf{Y})^{\text{H}}]=\text{E}[(\mathbf{H}+\mathbf{X}^{-1}\mathbf{Z})(\mathbf{H}+\bm {X}^{-1}\mathbf{Z})^{\text{H}}]\\ &=\text{E}[\mathbf{H}\widetilde{\mathbf{H}}^{\text{H}}+\mathbf{X}^{-1}\mathbf{Z}\widetilde{ \mathbf{H}}^{\text{H}}+\mathbf{H}\widetilde{\mathbf{Z}}^{\text{H}}(\mathbf{X}^{-1})^{\text{H} }+\mathbf{X}^{-1}\mathbf{Z}\widetilde{\mathbf{Z}}^{\text{H}}(\mathbf{X}^{-1})^{\text{H}}]\\ &=\text{E}[\mathbf{H}\widetilde{\mathbf{H}}^{\text{H}}]+\text{E}[\mathbf{X}^{-1}\mathbf{Z} \widetilde{\mathbf{Z}}^{\text{H}}(\mathbf{X}^{-1})^{\text{H}}]\\ &=\text{E}[\mathbf{H}\widetilde{\mathbf{H}}^{\text{H}}]+\sigma_{\text{n}}^{2}(\mathbf{X} \mathbf{X}^{\text{H}})^{-1}\\ &=\mathbf{R}_{\mathbf{H}\widetilde{\mathbf{H}}}+\sigma_{\text{n}}^{2}(\mathbf{X}\mathbf{X}^{ \text{H}})^{-1}\end{array} \tag{14}\]  
where \(\mathbf{Z}\) is AWGN. The MMSE channel estimation can be represented as:  
\[\hat{\mathbf{H}}_{\text{MMSE}}=\mathbf{R}_{\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}}(\bm {R}_{\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}}+\sigma_{\text{n}}^{2}(\mathbf{X}\mathbf{X} ^{\text{H}})^{-1})^{-1}\widetilde{\mathbf{H}}. \tag{15}\]  
In the case of all pilots, \(\mathbf{R}_{\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}}=\mathbf{R}_{\widetilde{\mathbf{H}} \widetilde{\mathbf{H}}}\). According to the designed orthogonal pilot pattern, for a particular transmit antenna pair, the channel coefficients of the pilot symbols can be represented as:  
\[\hat{\mathbf{H}}_{\text{MMSE}}^{\alpha,\beta}=\mathbf{R}_{\mathbf{H}\widetilde{\mathbf{H}}}( \mathbf{R}_{\mathbf{H}\widetilde{\mathbf{H}}}+\hat{\sigma}_{\text{n}}^{2}(\mathbf{X}_{\text{P} }\mathbf{X}_{\text{P}}^{\text{H}})^{-1})^{-1}\hat{\mathbf{H}}_{\text{LS}}^{\alpha, \beta}. \tag{16}\]