# Singular Value Decomposition Channel Estimation in STBC MIMO-OFDM System
## 2 System Model and STBC
### STBC and STBD  
Diversity technology is an important part of the multiple antennas technology. Diversity technology uses multiple antennas to receive or transmit signals carrying the same information, thereby improving the reliability of transmission. Diversity gains can be obtained by using the STBC technology at the transmitter. At the receiver, STBD only requires simple linear operation. In MIMO-OFDM systems, STBC is a key technology to improve the system performance. The Alamouti code is a complex orthogonal STBC specially designed for two transmit antennas [22]. According to the space-time coding scheme proposed by [PERSON], the symbols of the \(i\) frame and \(i+1\) frame on two transmit antennas are space-time coded in the frequency domain. Taking the first two frames of the OFDM symbols as an example, the model of the space-time encoder is shown in Figure 2.  
As shown in Figure 2, the input bits are modulated by the constellation modulation. Then, the modulated symbols are encoded in the space-time encoder. The space-time coded signals are transmitted from two antennas. After the IFFT operation, the frequency-domain OFDM symbols \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\) are transformed into time-domain OFDM symbols \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). At a given symbol period, two signals are simultaneously transmitted from two antennas. The signals transmitted from the first antenna are denoted by \(\mathbf{x}_{1}\) and from the second antenna by \(\mathbf{x}_{2}\). During the next symbol period, signal transmitted from the first antenna, and signal \(x_{1}^{*}\) is transmitted from the second antenna, where the superscript \((\cdot)^{*}\) denotes the conjugate operation.  
According to the Alamouti criterion [22], the OFDM symbols are sent out on the two transmit antennas, respectively. The Alamouti criterion between the \(i\) frame and \(i+1\) frame also can be shown as [22]:  
\[\mathbf{X}=\left[\begin{array}{cc}X_{i}(k)&-X_{i+1}^{*}(k)\\ X_{i+1}(k)&-X_{i}^{*}(k)\end{array}\right], \tag{3}\]  
where \(X_{i}(k)\) represents the frequency-domain data which transmits on the first antenna. It is located on the \(k\)-th subcarrier of the \(i\)-th OFDM symbol. \(X_{i+1}(k)\) is the frequency-domain data which transmits on the second antenna. According to the estimated channel frequency response (CFR), the OFDM symbol after STBD in the receiver can be shown as:  
\[\left[\begin{array}{c}\hat{\mathbf{X}}(i)\\ \hat{\mathbf{X}}(i+1)\end{array}\right]=\frac{\left[\begin{array}{cc}Y_{1}(i) \hat{H}_{1,1}^{*}(i)+Y_{1}^{*}(i+1)\hat{H}_{2,1}(i)+Y_{2}(i)\hat{H}_{1,1}^{*}( i)+Y_{2}^{*}(i+1)\hat{H}_{2,2}(i)\\ Y_{1}(i)\hat{H}_{2,1}^{*}(i)-Y_{1}^{*}(i+1)\hat{H}_{1,1}(i)+Y_{2}(i)\hat{H}_{2,2 }^{*}(i)-Y_{2}^{*}(i+1)\hat{H}_{1,2}(i)\\ \left|\hat{H}_{1,1}(i)\right|^{2}+\left|\hat{H}_{1,2}(i)\right|^{2}+\left| \hat{H}_{2,1}(i)\right|^{2}+\left|\hat{H}_{2,2}(i)\right|^{2}\end{array}\right]}, \tag{4}\]  
where \(\hat{\mathbf{X}}(i)\) and \(\hat{\mathbf{X}}(i+1)\) are the \(i\)-th and \(i+1\)-th STBD OFDM symbols, respectively. \(Y_{1}(i)\) is the \(i\)-th received symbol from the first antenna. \(Y_{2}(i)\) is the \(i\)-th received symbol from the second antenna. \(\hat{H}_{1,1}(i)\) represents the estimated CFR between the first transmit antenna and the first receive antenna. \(\hat{H}_{1,2}(i)\) represents the estimated CFR between the first transmit antenna and the second receive antenna. \(\hat{H}_{2,1}(i)\) represents the estimated CFR between the second transmit antenna and the first receive antenna. \(\hat{H}_{2,2}(i)\) represents the estimated CFR between the second transmit antenna and the second receive antenna.