# A Pilot-Pattern Based Algorithm for MIMO-OFDM Channel Estimation
## 3 Channel Estimation Algorithm Based on Space-Time Block-Code
### Analysis on the Algorithm for LS Channel Estimation
Meanwhile, suppose the increase of the time delay is much smaller than the interval of each OFDM sub-carrier, it can be concluded that \(H_{i}(n,2m)=H_{i}(n,2m+1)\). Now, defining the following equations:  
\[ \begin{array}{l}z_{1}(n,m)=\frac{r(n,2m)}{l_{2}(n,2m)}-\frac{r(n,2m+1)}{l_ {2}(n,2m+1)},\ z_{2}(n,m)=\frac{r(n,2m)}{l_{1}(n,2m)}-\frac{r(n,2m+1)}{l_{1}(n,2m+1)},\\ \overline{w}_{1}(n,m)=\frac{v(n,2m)}{l_{2}(n,2m)}-\frac{v(n,2m+1)}{l_{2}(n,2m+ 1)},\ \overline{w}_{2}(n,m)=\frac{w(n,2m)}{l_{1}(n,2m)}-\frac{w(n,2m+1)}{l_{1}(n,2m+1)}\end{array} \]  
it can be deduced that:  
\[ \begin{array}{l}z_{i}(n,m)=H_{i}(n,2m)v_{i}(n,m)+\overline{w}_{i}(n,m)\end{array}  \tag{11}\]  
where \(v_{1}(n,m)=\frac{t_{1}(n,2m)}{l_{2}(n,2m)}-\frac{t_{1}(n,2m+1)}{l_{2}(n,2m+1)}\) and \(v_{2}(n,m)=\frac{t_{2}(n,2m)}{t_{1}(n,2m)}-\frac{t_{2}(n,2m+1)}{l_{1}(n,2m+1)}\). It can be seen that the different transmission antennas are decoupled. The equivalent model of the transceiver signal is the one in which the received signal is only relevant to the raw transmitted signal. This estimation can be implemented through each transceiver antenna pair. The cost function of MSE is:\[C\Big{(}\{\hat{h}_{i}(n,l);\,i=1,2\}\Big{)}=\sum\limits_{k=0}^{M-1}\left|z_{i}(n,k )-\sum\limits_{l=0}^{K_{0}-1}\hat{h}_{i}(n,l)W_{M}^{kl}v_{i}(n,k)\right|^{2},\,M= K/2 \tag{12}\]  
It is necessary to regulate the following condition to minimize the cost function in Equation (12)  
\[\frac{\partial C\Big{(}\{\hat{h}_{i}(n,l)\}\Big{)}}{\partial\Big{(}\hat{h}_{i} (n,l_{0})\Big{)}}=0\]  
Then there exists the following equation according to the above condition:  
\[\sum\limits_{k=0}^{M-1}\left(z_{i}(n,k)-\sum\limits_{l=0}^{K_{0}-1}\hat{h}_{ i}(n,l)W_{M}^{kl}v_{i}(n,k)\right)W_{M}^{-kl_{0}}v_{i}^{*}(n,k)=0\]  
If we define \(p_{j}(n,j)=\sum\limits_{k=0}^{M-1}z_{j}(n,k)v_{j}^{*}(n,k)W_{M}^{-kl}\) and \(q_{i}(n,j)=\sum\limits_{k=0}^{M-1}|v_{i}(n,k)|^{2}W_{M}^{-kl}\), then \(\hat{\mathbf{h}}_{i}(n)\) can be expressed as:  
\[\hat{\mathbf{h}}_{i}(n)=\mathbf{Q}_{i}^{-1}(n)\mathbf{P}_{i}(n) \tag{13}\]  
where:  
\[\mathbf{Q}_{i}(n)=\left[ \begin{array}{cccc}q_{i}(n,0)&q_{i}(n,-1)&\cdots&q_{i}( n,-K_{0}+1)\\ q_{i}(n,1)&q_{i}(n,0)&\cdots&q_{i}(n,-K_{0}+2)\\ \vdots&\ddots&\ddots&\vdots\\ q_{i}(n,K_{0}-1)&q_{i}(n,K_{0}-2)&\cdots&q_{i}(n,0)\end{array} \right] \tag{14}\]  
Comparing matrix \(\mathbf{Q}\) in Equation (14) with that in Equation (10), it is obvious that the size of matrix \(\mathbf{Q}\) in Equation (14) and the necessary number of the Fast Fourier Transformation FFT) can be reduced by half. Note the decoupling method is used to get access to the matrix \(\mathbf{Q}\) with \(K_{0}\times K_{0}\), which indicates that if \(K_{0}\) is large in some extent, the computation overhead on matrix \(\mathbf{Q}\) is still large as well. If we ignore those \(h_{i}(n,l)\), which are much lower in power and use \(M\) most significant taps (MSTs) points and the corresponding MST method, a further simplification can be achieved with \(\overline{\mathbf{h}}(n)=\overline{\mathbf{Q}}^{-1}(n)\overline{\mathbf{P}}(n)\). In this case, solving inverse matrix is still essential. Now, expanding \(\overline{\mathbf{h}}(n)=\overline{\mathbf{Q}}^{-1}(n)\overline{\mathbf{P}}(n)\) as:  
\[\left[ \begin{array}{c}\hat{\mathbf{h}}_{1}(n)\\ \hat{\mathbf{h}}_{2}(n)\end{array} \right]=\left[ \begin{array}{cc}\mathbf{Q}_{11}(n) &\mathbf{Q}_{21}(n)\\ \mathbf{Q}_{12}(n)&\mathbf{Q}_{22}(n)\end{array} \right]^{-1}\left[ \begin{array}{c} \mathbf{p}_{1}(n)\\ \mathbf{p}_{2}(n)\end{array} \right] \tag{15}\]  
The usage of constant envelop modulation means \(|t_{i}(n,k)|=1\), \(|q_{ii}(n,l)|=K\delta(l)\), and \(\mathbf{Q}_{ii}(n)=KI\). In addition, if the specific training sequence that makes \(\mathbf{Q}_{ii}(n)=0\), \(i\
eq j\) is employed, then \(\mathbf{h}_{i}(n)=K^{-1}\mathbf{P}_{i}(n)\). The calculation of the inverse matrix of \(\mathbf{Q}\) is now changed to be a simple algebraic computation.