# Channel Estimation Based on Statistical Frames and Confidence Level in OFDM Systems
## 3 System Model
### IMMSE Channel Estimation Method  
Through the LS channel estimation in the OFDM receiver, it may not suppress the effect of noise efficiently in the time and frequency domains. The channel response values beyond the length of GI are set to zero to remove the impact of AWGN in this method.  
As shown in Figure 1, the CFR \(\hat{\mathbf{H}}_{\text{LS},i}(n)\) obtained by the LS channel estimation method is converted into time-domain CIR \(\hat{\mathbf{h}}_{\text{LS},i}(n)\) with \(n=0,1,2,\cdots,N-1\) by the IFFT operation. Assuming that \(L\) is the number of points of GI, the length of CIR should be no longer than that of the GI. To suppress the AWGN effectively, \(\hat{\mathbf{h}}_{\text{LS},i}(n)\) should be considered to be AWGN when \(n\geq L\). It can be given by:  
\[\hat{\mathbf{h}}_{\text{LS},i}(n)=\left\{\begin{array}{ll}\hat{\mathbf{h}}_{\text{LS },i}(n),&0\leq n<L,\\ 0,&L\leq n<N.\end{array}\right. \tag{6}\]  
The cross covariance of \(\hat{\mathbf{h}}_{i}(n)\) and \(\hat{\mathbf{y}}_{i}(n)\) is given by [11]:  
\[\mathbf{R}_{\hat{\mathbf{h}}_{i}(n)\hat{\mathbf{y}}_{i}(n)}=L\hat{\mathbf{\sigma}}_{h}^{2}\hat {\mathbf{x}}_{i}^{*}(n),\ 0\leq n<N, \tag{7}\]  
where \((\cdot)^{*}\) is the operation of complex conjugate transpose. The autocorrelation of \(\hat{\mathbf{y}}_{i}(n)\) is given by [11]:  
\[\mathbf{R}_{\hat{\mathbf{y}}_{i}(n)\hat{\mathbf{y}}_{i}(n)}=I\hat{\mathbf{\sigma}}_{h}^{2}| \hat{\mathbf{x}}_{i}(n)|^{2}+N\hat{\mathbf{\sigma}}_{n^{\prime}}^{2},\ 0\leq n<N, \tag{8}\]where \(I\) is the number of multipaths and \(N\) is the number of subcarriers. \(\hat{\mathbf{\sigma}}_{h}^{2}\) and \(\hat{\mathbf{\sigma}}_{n}^{2}\) are the estimated channel and noise variance, respectively. The estimated MMSE CIR \(\hat{\mathbf{h}}_{\text{MMSE},i}(n)\) could be represented as:  
\[\hat{\mathbf{h}}_{\text{MMSE},i}(n)=\frac{I\hat{\sigma}_{h}^{2}\hat{\mathbf{x}}_{i}^{*} (n)\hat{\mathbf{x}}_{i}(n)\hat{\mathbf{h}}_{\text{LS},i}(n)}{I\hat{\sigma}_{h}^{2}|\hat {\mathbf{x}}_{i}(n)|^{2}+N\hat{\sigma}_{n}^{2}},\ 0\leq n<N. \tag{9}\]  
In the dynamic multipath condition, it is difficult to differentiate \(\hat{\mathbf{h}}_{i}(n)\) and \(\hat{\mathbf{w}}_{i}(n)\). Suppose \(|\hat{\mathbf{x}}_{i}(n)|^{2}=1\). In the worst multipath condition, suppose \(\mathbf{\sigma}_{n}^{2}=\max(|\hat{\mathbf{h}}_{i}(n)|),\ 0\leq n<L\). Equation (9) can be represented as:  
\[\hat{\mathbf{h}}_{\text{MMSE},i}(n)=\frac{I\hat{\sigma}_{h}^{2}\hat{\mathbf{h}}_{ \text{LS},i}(n)}{I\hat{\sigma}_{h}^{2}+N\hat{\sigma}_{n}^{2}},\ 0\leq n<N. \tag{10}\]  
Suppose \(\hat{\sigma}_{h}^{2}=|\hat{\mathbf{h}}_{\text{LS},i}(n)|^{2}\). After suppressing the AWGN in time domain, the estimated CIR \(\hat{\mathbf{h}}_{\text{IMMSE},i}(n)\) can be expressed as [15]:  
\[\hat{\mathbf{h}}_{\text{IMMSE},i}(n)=\frac{\hat{\mathbf{h}}_{\text{LS},i}(n)\cdot| \hat{\mathbf{h}}_{\text{LS},i}(n)|^{2}}{\alpha\cdot|\hat{\mathbf{h}}_{\text{LS},i}(n) |^{2}+(1-\alpha)\cdot A_{i}^{2}},\ 0\leq n<N. \tag{11}\]  
In Equation (11), \(\alpha\) is the suppression factor, which is usually chosen in the value region of \([0.99,1]\). \(A_{i}=\underset{0\leq n<N}{\max}(|\hat{\mathbf{h}}_{\text{LS},i}(n)|)\).  
The disadvantage of the IMMSE method in time domain is that it suppresses some paths with small energy as the noise. The IMMSE algorithm involves the multiplication and division operations of \(N\) points, with large amount of computation and high complexity. Therefore, it is hard to implement in the hardware of the OFDM receiver.