# Singular Value Decomposition Channel Estimation in STBC MIMO-OFDM System
## 4 The Proposed SVD Channel Estimation Method  
LS and threshold methods presented in Section 3 can easily be implemented in OFDM systems. However, the OFDM symbols received by the receiver of the MIMO-OFDM system are the linear superposition of transmit symbols on each transmit antenna which can be presented as:  
\[\mathbf{Y}^{\beta}=\sum_{\alpha=1}^{N_{\text{T}}}\mathbf{H}^{\alpha,\beta}\mathbf{X}^{ \alpha,\beta}, \tag{7}\]  
where the \(N\times 1\) vector \(\mathbf{Y}^{\beta}\) is the received OFDM symbol in the \(\beta\)-th receive antenna. \(N_{\text{T}}\) is the number of the transmit antennas. The \(N\times N\) diagonal matrix \(\mathbf{H}^{\alpha,\beta}\) is the CFR between the \(\alpha\)-th transmit antenna and the \(\beta\)-th receive antenna. The \(N\times 1\) vector \(\mathbf{X}^{\alpha,\beta}\) is the transmitting symbol between the \(\alpha\)-th transmit antenna and the \(\beta\)-th receive antenna. The pilot symbols can be orthogonal to eliminate the interference between the transmit antennas and receive antennas. Hence, this paper designs an orthogonal comb-type pilot pattern which is shown in Figure 4.  
Figure 3: Threshold channel estimation method. As shown in Figure 4, the green filling block represents transmitted data symbols. The yellow filling block represents zero, and the red filling block represents the transmitted pilot symbol. The pilot positions of the two antennas are orthogonal to each other, and the pilot symbols are orthogonal to each other. According to this pilot pattern, for a particular transmit antenna pair, the pilot symbols cannot be affected by other transmit antenna symbols. As a result, MIMO channel estimation can be converted into a number of independent single input and single output channel estimation problems. The estimated CFR \(\mathbf{H}_{\text{LS}}^{\alpha,\beta}(k_{\text{P}})\) between the \(\alpha\)-th transmit antenna and the \(\beta\)-th receive antenna according to the LS method which can be represented as:  
\[\hat{\mathbf{H}}_{\text{LS}}^{\alpha,\beta}(k_{\text{P}})=\frac{\mathbf{Y}^{\alpha, \beta}(k_{\text{P}})}{\mathbf{X}^{\alpha}(k_{\text{P}})}, \tag{8}\]  
where \(\mathbf{Y}^{\alpha,\beta}(k_{\text{P}})\) is the received pilot symbol on the \(k_{\text{P}}\)-th subcarrier between the \(\alpha\)-th transmit antenna and the \(\beta\)-th receive antenna. \(\mathbf{X}^{\alpha}(k_{\text{P}})\) is the transmitted pilot symbol on the \(k_{\text{P}}\)-th subcarrier of the \(\alpha\)-th transmit antenna. According to the spline interpolation, the CFR \(\hat{\mathbf{H}}_{\text{LS}}^{\alpha,\beta}\) in data positions can be obtained. In the threshold channel estimation method, it is difficult to choose the appropriate threshold to suppress the noise in channel coefficients, which restrains the application in real scenario. The MMSE channel estimation method has better performance than the LS and threshold methods since it is optimum in minimizing the mean square error (MSE) of the channel estimates in the presence of AWGN. According to the pilot sequence designed in Figure 4, the MMSE channel estimation can be directly applied in STBC MIMO-OFDM systems without increasing the computational complexity. However, the MMSE channel estimation method requires the inverse operation of the channel correlation matrix, which has a large amount of computation. To further reduce the calculation complexity, the SVD method is used to decompose the channel autocorrelation matrix which avoids the matrix inversion operation. In Appendix A, the MMSE channel estimation method in STBC MIMO-OFDM systems can be represented as:  
\[\hat{\mathbf{H}}_{\text{MMSE}}^{\alpha,\beta}=\mathbf{R}_{\text{HH}}(\mathbf{R}_{\text{HH}} +\hat{\sigma}_{\text{n}}^{2}(\mathbf{X}_{\text{P}}\mathbf{X}_{\text{P}}^{\text{H}})^{ -1})^{-1}\hat{\mathbf{H}}_{\text{LS}}^{\alpha,\beta}, \tag{9}\]  
where \(\mathbf{R}_{\text{HH}}=\text{E}(\mathbf{H}\mathbf{H}^{\text{H}})\) is the \(N\times N\) channel autocorrelation matrix. \(\mathbf{X}_{\text{P}}\) is the transmitted pilot symbol.