# Joint Channel Estimation Algorithm Based on DFT and DWT
## Appendix B Noise Power Boundary Analysis  
For a single estimated \(h_{\text{V}}[n]\) and a given threshold \(T\), among the samples with the same or similar channel component energies, those samples with low power noise superimposed are more likely to be judged as noise samples and are set to zero. Therefore, it is reasonable to consider that the total noise power \(\sum\limits_{n\in\Theta}|w[n]|^{2}\) on the selected nulling samples (assuming \(K\) samples) is not greater than \(K\) times the average noise power; that is,  
\[\sum\limits_{n\in\Theta}|w[n]|^{2}\leq K\sigma_{\text{s-wt}}^{2}=\frac{KN_{p} }{N^{2}}\sigma_{\text{s-w}f}^{2}, \tag{16}\]  
where \(\sigma_{\text{s-w}f}^{2}\) is the average power of the current OFDM symbol frequency domain noise, and \(\widehat{\sigma}_{\text{s-wt}}^{2}\) is the average power of the current OFDM symbol time-domain noise in the presence of a virtual subcarrier. On the other hand, \(\sum\limits_{n\in\Theta}\left|w[n]\right|^{2}\) will not be less than the total noise power on the \(K\) samples with the lowest noise power among \(N\) samples; that is,  
\[\sum\limits_{n\in\Omega_{K}}\left|w[n]\right|^{2}\leq\sum\limits_{n\in\Theta} \left|w[n]\right|^{2}, \tag{11}\]  
where \(\Omega_{K}\) is the set of \(K\) samples with the lowest noise power. It is assumed that the maximal noise power superimposed on these \(K\) samples is \(Z_{\text{max}}\), and \(Z_{\text{max}}=max\Big{\{}\left|w[n]\right|^{2}\Big{\}}\), \(n\in\Omega_{K}\). Then, for all \(N\) estimated samples in each estimated CIR, the probability of the occurrence of estimated paths with a noise power value of less than or equal to \(Z_{\text{max}}\) is \(\frac{K}{N}\). It can also be considered that, for the noise sequence \(w[n]\), the probability that its power \(\left|w[n]\right|^{2}\) is less than or equal to \(Z_{\text{max}}\) is \(\frac{K}{N}\). To simplify this description, denote \(\left|w[n]\right|^{2}\) as \(z\). Then, we have  
\[P(z\leq Z_{\text{max}})=\frac{K}{N}. \tag{12}\]  
Since \(w[n]\) can be regarded as a complex random variable with zero mean and variance \(\widetilde{\sigma}_{wt}^{2}\), \(z\) obeys a chi-square distribution with 2 degrees of freedom, whose probability density function and cumulative distribution function are expressed as [24]  
\[f(z)=\frac{1}{\delta_{wt}^{2}}\text{e}^{-\frac{z}{\delta_{wt}^{2}}}. \tag{13}\]  
\[F(z)=1-\text{e}^{-\frac{z}{\delta_{wt}^{2}}}. \tag{14}\]  
Therefore, we get  
\[P(z\leq Z_{\text{max}})=\int_{0}^{Z_{\text{max}}}f(z)\text{d}z=1-\text{e}^{- \frac{Z_{\text{max}}}{\delta_{wt}^{2}}}. \tag{15}\]  
From Equations (12) and (15) we can obtain \(1-\text{e}^{-\frac{Z_{\text{max}}}{\delta_{wt}^{2}}}=\frac{K}{N}\); that is,  
\[Z_{\text{max}}=-\delta_{wt}^{2}\text{ln}\bigg{(}1-\frac{K}{N}\bigg{)}. \tag{16}\]  
A mean of \(z\) that is less than or equal to \(Z_{\text{max}}\) can be expressed as  
\[E(z|z\leq Z_{\text{max}})=\int_{0}^{+\infty}zf(z|z\leq Z_{\text{max}})\text{ d}z. \tag{17}\]  
where \(f(z|z\leq Z_{\text{max}})\) can be obtained by the derivation of \(F(z|z\leq Z_{\text{max}})\). If \(z>Z_{\text{max}}\), \(F(z,z\leq Z_{\text{max}})=F(z>Z_{\text{max}},z\leq Z_{\text{max}})=0\). If \(z\leq Z_{\text{max}}\), \(F(z,z\leq Z_{\text{max}})=F(z\leq Z_{\text{max}},z\leq Z_{\text{max}})=F(z)\). Therefore, we have  
\[F(z,z\leq Z_{\text{max}})=\left\{ \begin{array}{cc}1-\text{e}^{-\frac{z}{ \delta_{wt}^{2}}},&z\leq Z_{\text{max}},\\ 0,&z>Z_{\text{max}}.\end{array} \right. \tag{18}\]From the Bayesian equation, we can obtain  
\[ \begin{split} F(z|z&\leq Z_{\max})=\frac{F(z,z\leq Z_{ \max})}{F(z\leq Z_{\max})}=\frac{F(z,z\leq Z_{\max})}{P(z\leq Z_{\max})}\\ &=\left\{\begin{array}{cc}\frac{1-\frac{-\frac{z}{\beta_{\rm int }^{2}}}{\sigma_{\rm int}^{2}}}{1-\frac{-\frac{z}{\beta_{\rm int}}}{\sigma_{ \rm int}^{2}}},&z\leq Z_{\max}\\ 0,&z>Z_{\max}\end{array}\right.\\ &=\left\{\begin{array}{cc}\frac{N}{K}\bigg{(}1-\mathrm{e}^{-\frac{z}{\beta_{ \rm int}^{2}}}\bigg{)},&z\leq Z_{\max},\\ 0,&z>Z_{\max}.\end{array}\right.\end{split}