# Joint Channel Estimation Algorithm Based on DFT and DWT
## Appendix A MSE of DFT-Based Channel Estimation in OFDM Systems with Virtual Subcarriers  
Let \(\emptyset\) be the set of zero-set samples in \(\hat{h}_{\text{V}}[n]\), and let the number of samples in the set be \(K\). The CIR after zero-set samples can be expressed as  
\[\hat{h}_{\text{V-zero}}[n]=\left\{ \begin{array}{cc}0,&n\in\emptyset,\\ \hat{h}_{\text{V}}[n],&\text{otherwise}.\end{array} \right. \tag{1}\]  
when virtual subcarriers exist, the MSE of \(k\)th subcarrier of the plus-window DFT algorithm can be approximated as  
\[ \begin{split}&\text{MSE}_{\text{V-W-DFT}}[k]=E\bigg{[}\Big{|} \text{DFT}_{N}\Big{\{}h_{\text{V}}[n]-\hat{h}_{\text{V-zero}}[n]\Big{\}} \Big{|}^{2}\bigg{]}\\ &\qquad=E\Bigg{[}\bigg{|}\text{DFT}_{N}\Bigg{\{}\sum\limits_{n\in \mathcal{O}}h_{\text{V}}[n]-\sum\limits_{n=0,n\
eq\mathcal{O}}^{N-1}\bar{w} [n]\bigg{\}}\bigg{|}^{2}\bigg{]}\\ &\qquad=E\Bigg{[}\bigg{|}\sum\limits_{n\in\mathcal{O}}h_{\text{V}}[n]e^{- \frac{2\pi kn}{N}}\bigg{|}^{2}\bigg{]}+E\Bigg{[}\bigg{|}\sum\limits_{n=0,n \
eq\mathcal{O}}^{N-1}\bar{w}[n]e^{-\frac{2\pi kn}{N}}\bigg{|}^{2}\bigg{]}, \end{split}  \tag{2}\]  
where \(\text{DFT}_{N}\{\cdot\}\) is \(N\)-point DFT, and  
\[ \begin{split}& E\Bigg{[}\bigg{|}\sum\limits_{n\in\mathcal{O}}h_{ \text{V}}[n]e^{-\frac{2\pi kn}{N}}\bigg{|}^{2}\bigg{]}=E\Bigg{[}\sum\limits_{n _{1}\in\mathcal{O}}h_{\text{V}}[n_{1}]e^{-\frac{2\pi kn_{1}}{N}}\sum\limits_ {n_{2}\in\mathcal{O}}h_{\text{V}}^{*}[n]e^{\frac{2\pi kn_{2}}{N}}\Bigg{]}\\ &=E\Bigg{[}\sum\limits_{n\in\mathcal{O}}|h_{\text{V}}[n]|^{2}+ \sum\limits_{n_{1}\in\mathcal{O}}\sum\limits_{n_{2}\in\mathcal{O},n_{2}\
eq n _{1}}h_{\text{V}}[n_{1}]h_{\text{V}}^{*}[n_{2}]e^{\frac{-2\pi kn(n_{1}-n_{2} )}{N}}\Bigg{]}\\ &=\sum\limits_{n\in\mathcal{O}}E\Big{[}\big{|}h_{\text{V}}[n] \big{|}^{2}\Big{]}+\sum\limits_{n_{1}\in\mathcal{O}}\sum\limits_{n_{2}\in \mathcal{O},n_{2}\
eq n_{1}}e^{\frac{-2\pi kn(n_{1}-n_{2})}{N}}E\big{[}h_{\text{ V}}[n_{1}]h_{\text{V}}^{*}[n_{2}]\big{]}.\end{split}  \tag{3}\]Since each path of the channel is independent of the others, when \(n_{2}\
eq n_{1}\), there is \(E\big{[}h[n_{1}]h^{*}[n_{2}]\big{]}=0\), so  
\[ \begin{array}{l}E\left[\left|\sum\limits_{n\in\mathcal{O}}h_{\text{V}}[n]e^{- \frac{2\pi\text{ln}}{N}}\right|^{2}\right]=\sum\limits_{n\in\mathcal{O}}E\Big{[} \big{|}h_{\text{V}}[n]\big{|}^{2}\Big{]}+\\ \sum\limits_{n_{1}\in\mathcal{O}}\sum\limits_{n_{2}\in\mathcal{O},n_{2}\
eq n_ {1}}e^{-\frac{2\pi\text{ln}(n_{1}-n_{2})}{N}}\sum\limits_{n=0}^{N-1}r_{P}[[n_ {1}-n]]_{N}r_{P}^{*}[[n_{2}-n]]_{N}E\Big{[}\big{|}h[n]\big{|}^{2}\Big{]}.\end{array}  \tag{10}\]  
The second term to the right of the medium sign in Equation (10) reflects the correlation between the channel component \(h_{\text{V}}[n]\) due to the virtual subcarriers. When the number of virtual subcarriers is small, this term has a small value relative to the power of the channel component and can be neglected. Therefore, the above equation can be approximated as  
\[E\left[\left|\sum\limits_{n\in\mathcal{O}}h_{\text{V}}[n]e^{-\frac{2\pi\text{ ln}}{N}}\right|^{2}\right]\approx\sum\limits_{n\in\mathcal{O}}E\Big{[}\big{|}h_{ \text{V}}[n]\big{|}^{2}\Big{]}. \tag{11}\]  
In addition, in Equation (11)  
\[E\left[\left|\sum\limits_{n=0,n\
otin\mathcal{O}_{2}}\tilde{w}[n]e^{-\frac{2 \pi\text{ln}}{N}}\right|^{2}\right]=\sum\limits_{n_{1}=0,n_{1}\
otin\mathcal{O }_{2}}^{N-1}\sum\limits_{n_{2}=0,n_{2}\
otin\mathcal{O}_{2}}^{N-1}e^{-\frac{2 \pi\text{ln}(n_{1}-n_{2})}{N}}E[\tilde{w}[n_{1}]\tilde{w}^{*}[n_{2}]]. \tag{12}\]  
when the number of virtual subcarriers is small, \(\tilde{w}[n_{1}]\) and \(\tilde{w}[n_{2}]\) can be assumed to be independent of each other if \(n_{1}\
eq n_{2}\). Then,  
\[E[\tilde{w}[n_{1}]\tilde{w}^{*}[n_{2}]]\approx\left\{ \begin{array}{cc}E\big{[} \big{|}\tilde{w}[n]\big{|}^{2}\big{]},&n=n_{1}=n_{2},\\ 0,&\text{otherwise}.\end{array} \right.