# Channel Estimation Based on Statistical Frames and Confidence Level in OFDM Systems
## 5 Channel Estimation Based on Statistical Frames and the Confidence Level  
The confidence level shows that the true value of the estimated CIR has a certain probability to fall around the measurement interval. The confidence level gives the credibility of the measured values of the measured CIRs. This probability is called the confidence level. The paper presents an OFDM channel estimation method based on statistical frames and the confidence level, it has good performances in both static and dynamic multipath channels. The method is based on the statistical frames \(T\) and confidence level \(\mathbf{K}_{i}(n)\). For every \(T\) consecutive frames, the channel estimation results are counted once. The number of real and imaginary parts \(\hat{\mathbf{h}}_{re,i}(n)\), \(\hat{\mathbf{h}}_{im,i}(n)\) is counted in the intervals of \((-\infty,0)\) and \((0,+\infty)\). Assume the number of times that \(\hat{\mathbf{h}}_{re,i}(n)\), which occurs in the intervals of \((-\infty,0)\) and \((0,+\infty)\), is \(\mathbf{s}_{i1}\), \(\mathbf{s}_{i2}\). The number of times that \(\hat{\mathbf{h}}_{im,i}(n)\), which occurs in the intervals of \((-\infty,0)\) and \((0,+\infty)\), is \(\mathbf{s}_{i3}\) and \(\mathbf{s}_{i4}\). The maximum number of \(\hat{\mathbf{h}}_{re,i}(n)\), which occurs in the intervals of \((-\infty,0)\) and \((0,+\infty)\), is denoted as \(\mathbf{s}_{re}\). The maximum number of \(\hat{\mathbf{h}}_{im,i}(n)\), which occurs in the intervals of \((-\infty,0)\) and \((0,+\infty)\), is denoted as \(\mathbf{s}_{im}\). \(\mathbf{s}_{re}\) and \(\mathbf{s}_{im}\) can be represented as:  
\[\left\{ \begin{array}{l}\mathbf{s}_{re}=\max[\mathbf{s}_{i1},\mathbf{s}_{i2}],\\ \mathbf{s}_{im}=\max[\mathbf{s}_{i3},\mathbf{s}_{i4}].\end{array} \right. \tag{17}\]  
The PDF of noise obeys a Gaussian distribution. The PDF of the Gaussian distribution of noise under different statistical frames is represented in Figure 2. The literature [15] represents the figure of the PDF of the real parts of the path and the noise. The disadvantage is that it does not consider the size of the statistical frames, which could affect the probability of the noise distribution. When the statistical frames \(T\) increase from five to thirteen, the probability that \(p(\hat{\mathbf{h}}_{\text{LS},i}(n))\) occurs in the positive and negative intervals may increase. According to the PDF of the Gaussian distribution, if \(\hat{\mathbf{h}}_{\text{LS},i}(n)\) is noise, the number of its real part or imaginary part appears in the positive and negative intervals nearly equal, that is \(\mathbf{s}_{re}\approx T/2\). If \(\hat{\mathbf{h}}_{\text{LS},i}(n)\) is the multipath, the number of the real part or the imaginary part appearing in the statistical interval is approximately equal to the statistical frame number, that is \(\mathbf{s}_{re}\approx T\). Therefore, the more \(\mathbf{s}_{re}\) is closer to \(T\), the higher the confidence level of \(\hat{\mathbf{h}}_{\text{LS},i}(n)\) is; the more \(\mathbf{s}_{re}\) is closer to \(T/2\), the lower the confidence level of the \(\hat{\mathbf{h}}_{\text{LS},i}(n)\) is. The confidence level \(\mathbf{K}_{i}(n)\) can also be obtained by the soft decision method, which can be represented by:  
\[\mathbf{K}_{i}(n)=\left\{ \begin{array}{l}1,\quad\mathbf{s}_{iM}=T\\ \beta_{1},\ \mathbf{s}_{iM}=T-1\\ \beta_{2},\ \mathbf{s}_{iM}=T-2\\ \vdots,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\\ 0,\ \ \mathbf{s}_{iM}=0\end{array} \right. \tag{18}\]  
where \(\beta_{1}\), \(\beta_{2}\in(0,1)\) are the ranges of confidence level \(\mathbf{K}_{i}(n)\). Take the maximum of the four values of \(\mathbf{s}_{re}\) and \(\mathbf{s}_{im}\) is \(\mathbf{s}_{iM}\). \(\mathbf{s}_{iM}\) can be represented as:  
\[\mathbf{s}_{iM}=\max[\mathbf{s}_{re},\mathbf{s}_{im}]. \tag{19}\]  
Compare \(\mathbf{s}_{iM}\) with statistical frames \(T\). To simplify the calculation of the confidence level, \(\mathbf{K}_{i}(n)\) is obtained by using the hard decision method.