# A Pilot-Pattern Based Algorithm for MIMO-OFDM Channel Estimation
## 4 Improved Channel Estimation Algorithm Based on STBC  
Since the use of the single-symbol pilot needs the method of interpolation to obtain the channel response at all of the frequency points, except the pilot, interpolation error is inevitably introduced. In particular, if there is a large time delay, the error can never be ignored. Although this problem can be fixed by using the double-symbol pilot, the challenge of large computational overhead must be seriously considered. In these regards, figuring out novel algorithms to achieve an optimized performance trade-off is highly appreciated.  
Since the double-symbol pilot is turned into a special case of STBC when all of the pilot signal are real values, it is possible to take other kinds of special cases of STBC into consideration. Inspired by the concept that the optimal training sequence can use a diagonal matrix to avoid complicated computations on matrix inversion, the novel algorithm herein will be developed based on the idea that the algorithm should have the same pilot as the double-symbol scheme. Then the pilot number will be twice that of the single-symbol scheme, and be half that of the double-symbol scheme. The pilot is then configured as:  
\[\left[\begin{array}{cc}p&0\\ 0&p\end{array}\right]\]  
where \(p\) is a constant that represents the value of the transmitted pilot.  
When there are two antenna transmitting pilots, all of real values, in the crossover way, the expression of this system will be:  
\[\left[\begin{array}{cc}p&0\\ 0&p\end{array}\right]\left[\begin{array}{cc}H_{11}&H_{12}\\ H_{21}&H_{22}\end{array}\right]+\left[\begin{array}{cc}n_{11}&n_{12}\\ n_{21}&n_{22}\end{array}\right]=\left[\begin{array}{cc}r_{11}&r_{12}\\ r_{21}&r_{22}\end{array}\right]\] \[\Rightarrow\left[\begin{array}{cc}\hat{H}_{11\_LS}&\hat{H}_{12\_ LS}\\ \hat{H}_{21\_LS}&\hat{H}_{22\_LS}\end{array}\right]=\left[\begin{array}{cc}1/p&0\\ 0&1/p\end{array}\right]\left[\begin{array}{cc}r_{11}&r_{12}\\ r_{21}&r_{22}\end{array}\right]=\left[\begin{array}{cc}r_{11}/p&r_{12}/p\\ r_{21}/p&r_{22}/p\end{array}\right]\] \[\Rightarrow\hat{H}_{\vec{U}\_MMSE}=R_{HH}\big{(}R_{HH}+\frac{\beta} {SNR}I\big{)}^{-1}\hat{H}_{\vec{U}\_LS}\]In which \(\left[\begin{array}{cc}p&0\\ 0&p\end{array}\right]\) is the pilot matrix, \(\left[\begin{array}{cc}H_{11}&H_{12}\\ H_{21}&H_{22}\end{array}\right]\) is the transfer function matrix of the channels, \(\left[\begin{array}{cc}n_{11}&n_{12}\\ n_{21}&n_{22}\end{array}\right]\) is the matrix of additive white Gaussian noise, \(\left[\begin{array}{cc}r_{11}&r_{12}\\ r_{21}&r_{22}\end{array}\right]\) is the received signal matrix, \(\left[\begin{array}{cc}\hat{H}_{11\_LS}&\hat{H}_{12\_LS}\\ \hat{H}_{21\_LS}&\hat{H}_{22\_LS}\end{array}\right]\) is the estimation of \(\left[\begin{array}{cc}H_{11}&H_{12}\\ H_{21}&H_{22}\end{array}\right]\) on the basis of method LS, \(R_{HH}\) represents the autocorrelation matrix of matrix \(H\), \(I\) is a unit matrix, \(\beta\) is a constant dependent on the signal constellation, and \(SNR\) is the signal-to-noise ratio.  
It is noted that both the performance and computational overhead of LS are in between those of the single-symbol scheme and the double-symbol scheme [21].