# 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  
For spatial-temporal decoding, Viterbi decoding is commonly used integrating with the matrix parameter \(\left\|\mathbf{r}(n,k)-\hat{\mathbf{H}}(n,k)\hat{\mathbf{t}}(n,k)\right\|^{2}\), where \(\left\|\cdot\right\|^{2}\) represents the norm of the Euclidean. \(r(n,k)\), \(H(n,k)\), and \(t(n,k)\) can be expressed as:  
\[\mathbf{r}(n,k)=\left[ \begin{array}{c}r_{1}(n,k)\\ r_{2}(n,k)\end{array} \right],\ \hat{\mathbf{H}}(n,k)=\left[ \begin{array}{cc}\hat{H}_{11}(n,k)& \hat{H}_{21}(n,k)\\ \hat{H}_{12}(n,k)&\hat{H}_{22}(n,k)\end{array} \right],\ \hat{\mathbf{t}}(n,k)=\left[ \begin{array}{c}t_{1}(n,k)\\ t_{2}(n,k)\end{array} \right]\]  
Since channel parameters are included in the elements in above matrix, whether the parameters are correct or not unavoidably influence the performance of the spatial-temporal codec. If the channels exist in various pairs of antennas that are located in the transmitter and receiver, respectively, are independent of each other, the channel estimation for each antenna is independent as well. Therefore, it is feasible to issue the channel estimation for each antenna on the receiver side, which can be formulated as:  
\[r(n,k)=\sum\limits_{i=1}^{2}H_{i}(n,k)t_{i}(n,k)+w(n,k),\ k=0,1,\ldots,K-1 \tag{6}\]  
If the transmitted signal \(t_{i}(n,k)\) is substituted by a training sequence, the estimation on \(h_{i}(n,l)\) can be performed by minimizing the cost function of the mean square error (MSE) as written by:  
\[C\Big{(}\{\hat{h}_  
For the above formula, \(\hat{h}_{i}(n,l)\) can be computed through solving the following equation:  
\[\frac{\partial C\Big{(}\{\hat{h}_{i}(n,l)\}\Big{)}}{\partial\Big{(}\hat{h}_{i}(n,l_{0})\Big{)}}=0\]  
and:  
\[\sum\limits_{k=0}^{K-1}\Bigg{(}r(n,k)-\sum\limits_{i=1}^{2}\sum\limits_{l=0}^{ K_{0}-1}\hat{h}_{i}(n,l)W_{K}^{kl}t_{i}(n,k)\Bigg{)}W_{K}^{-kl_{0}}t_{i}^{*}(n,k)=0 \tag{8}\]  
where \(j=1,2\), \(l_{0}=0,1,\ldots,K_{0}-1\). Now, given \(p_{j}(n,j)=\sum\limits_{k=0}^{K-1}r(n,k)t_{j}^{*}(n,k)W_{K}^{-kl}\) and \(q_{ij}(n,l)=\sum\limits_{k=0}^{K-1}t_{i}(n,k)t_{j}^{*}(n,k)W_{K}^{-kl}\), then there exist the following equations:  
\[p_{j}(n,l_{0})=\sum\limits_{i=1}^{2}\sum\limits_{l=0}^{K_{0}-1}\hat{h}_{i}(n, l)q_{ij}(n,l_{0}-l),\ \hat{\mathbf{h}}_{i}(n)=\mathbf{Q}^{-1}(n)\mathbf{P}(n) \tag{9}\]  
where:  
\[ \begin{array}{l}\hat{\mathbf{h}}(n)=\left[\begin{array}{c}\hat{\mathbf{h}}_{1}(n) \\ \hat{\mathbf{h}}_{2}(n)\\ \end{array}\right]\!,\ \hat{\mathbf{h}}_{i}(n)=\left[\begin{array}{cccc}\hat{h}_{i}(n,0)&\hat{h}_ {i}(n,1)&\cdots&\hat{h}_{i}(n,K_{0}-1)\\ \end{array}\right]^{\mathrm{T}}\\ \mathbf{P}(n)=\left[\begin{array}{c}\mathbf{P}_{1}(n)\\ \mathbf{P}_{2}(n)\\ \end{array}\right]\!,\ \mathbf{P}_{i}(n)=\left[\begin{array}{cccc}P_{i}(n,0)&P_{i}(n,1)& \cdots&P_{i}(n,K_{0}-1)\\ \end{array}\right]^{\mathrm{T}}\\ \mathbf{Q}(n)=\left[\begin{array}{cccc}\mathbf{Q}_{11}(n)&\mathbf{Q}_{21}(n)\\ \mathbf{Q}_{12}(n)&\mathbf{Q}_{22}(n)\\ \end{array}\right]\!,\\ \mathbf{Q}_{ij}(n)=\left[\begin{array}{cccc}q_{ij}(n,0)&q_{ij}(n,-1)&\cdots&q_{ ij}(n,-K_{0}+1)\\ q_{ij}(n,1)&q_{ij}(n,0)&\cdots&q_{ij}(n,-K_{0}+2)\\ \vdots&\ddots&\ddots&\vdots\\ q_{ij}(n,K_{0}-1)&q_{ij}(n,K_{0}-2)&\cdots&q_{ij}(n,0)\\ \end{array}\right]\end{array}  \tag{10}\]  
It can be seen from Equation (9) that the implementation of channel estimation needs to calculate the inverse matrix of \(\mathbf{Q}\). However, the larger the time delay, the larger \(K_{0}\), which results in the fact that the overhead of computing the inverse matrix of \(\mathbf{Q}\) will also increase with the value of \(K_{0}\). Thus, how to decrease the volume of \(\mathbf{Q}\) and how to simply the computation on solving the inverse matrix of \(\mathbf{Q}\) becomes much more important in the process of channel estimation. When the time delay is not particularly large, the response of a channel is relevant to each sub-carrier located in this channel.