# Singular Value Decomposition Channel Estimation in STBC MIMO-OFDM System
## 2 System Model and STBC  
### STBC MIMO-OFDM System Model  
This paper considers a system with two transmit and receive antennas, as shown in Figure 1.  
For a system with \(N_{\mathrm{T}}>2\) transmit antennas and \(N_{\mathrm{R}}>2\) receive antennas, the new orthogonal pilot sequences need to be designed to suppress the interference from other transmit antennas.  
Figure 1: System model of the space-time block coding multiple-input multiple-output orthogonal frequency division multiplexing (STBC MIMO-OFDM) system.  
The space-time block code should be designed for \(N_{\rm T}>2\) transmit antennas. The system has \(N\) subcarriers in an OFDM block. At the transmitter side, the input bits are modulated in the constellation block. Then, the modulation symbols are sent to the space-time encoder. According to the principle of Alamouti criterion [22], the space-time coded signals are allocated to two antennas. The pilots are inserted into the OFDM symbols transmitted on two antennas, respectively. After the inverse fast Fourier transform (IFFT) operation, the data symbols \(\mathbf{X}_{i}^{\alpha}(k)\) are transformed into the time domain. The time-domain OFDM symbols \(\mathbf{x}_{i}^{\alpha}(k)\) can be shown as:  
\[\mathbf{x}_{i}^{\alpha}(k)=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}\mathbf{X}_{i}^{\alpha} \mathrm{e}^{\mathrm{i}2\pi\Omega n/N},k=0,1,\cdots,N-1, \tag{1}\]  
where \(\alpha=1,2\) is the transmit antenna number and \(k\) represents the index of subcarriers. The comb-type pilots in two antennas are different and orthogonal. Then, the CP is inserted into the time-domain OFDM symbols in order to reduce the ISI. After the parallel to serial (P/S) block, the space-time block coded signals are transmitted from two antennas, respectively. In the process of channel propagation, the transmitted symbol will suffer from the multipath fading and AWGN. The channel environment will affect the correct receive of the receiver.  
At the receiver, the received serial OFDM symbols are transformed into parallel in the serial to parallel (S/P) block. Then, the CP is removed from the parallel OFDM symbols. After removing the CP, the time-domain received OFDM symbols \(\mathbf{y}_{i}^{\beta}(k)\) are transformed into frequency domain in the fast Fourier transform (FFT) block. \(\beta=1,2\) is the receive antenna number. The received symbols in the frequency domain can be written in vector form as:  
\[\mathbf{Y}_{i}=\mathbf{H}_{i}\mathbf{X}_{i}+\mathbf{W}_{i}, \tag{2}\]  
where \(\mathbf{X}_{i}=[X_{i}^{1},X_{i}^{2}]^{\rm T}\) are the transmitted OFDM symbols of the \(i\)-th signal frame on two transmit antennas. \(\mathbf{Y}_{i}=[Y_{i}^{1},Y_{i}^{2}]^{\rm T}\) and \(\mathbf{W}_{i}=[W_{i}^{1},W_{i}^{2}]^{\rm T}\) are the received OFDM symbols and AWGN on two receive antennas, respectively. \(\mathbf{H}_{i}\) denotes the \(2\times 2\) channel matrix. Then, the comb-type pilots are extracted to estimate the CSI. Using the estimated CSI, the frequency-domain symbols are decoded in the space-time decoder block. The signals at multiple receive antennas are decoded using CSI provided by the channel estimator. Therefore, channel estimation is a critical module for STBC MIMO-OFDM systems. Finally, OFDM symbols are demodulated into bits data in the constellation demodulation block.