# Joint Channel Estimation Algorithm Based on DFT and DWT
## 3 DFT-Based Channel Estimation Algorithm  
### Traditional DFT-Based Channel Estimation Algorithm  
The DFT-based channel estimation algorithm takes advantage of the concentration of channel power in a few samples in the CIR for noise reduction to further improve the estimation accuracy. The conventional DFT-based channel estimation algorithm first converts the CFR estimated by the LS algorithm to the time-domain via IDFT to obtain the estimated time-domain channel impulse response, i.e., [11]:  
\[\hat{h}_{\text{LS}}[n]=\frac{1}{N}\sum\limits_{k=0}^{N-1}\hat{H}_{\text{LS}}[k ]exp\left(\frac{j2\pi kn}{N}\right)=h_{\text{LS}}[n]+w[n], \tag{9}\]  
where \(h_{\text{LS}}[n]\) denotes the channel component on the CIR, and \(w[n]\) is the time-domain complex Gaussian white noise. In the absence of virtual subcarriers, the channel components exist only within the CP length of \(\hat{h}_{\text{LS}}[n]\), so all samples outside the CP length can be considered noise and are removed by setting their values to zero. This process can be regarded as adding a window to the whole \(\hat{h}_{\text{LS}}[n]\). The \(\hat{h}_{\text{LS}}[n]\) after windowing can be expressed as  
\[\hat{h}_{\text{W-DFT}}[n]=\hat{h}_{\text{LS}}[n]R_{\text{DFT}}[n]. \tag{10}\]  
\[R_{\text{DFT}}[n]=\left\{ \begin{array}{ll}1,&n=0,1,\cdots,N_{\text{CP}}-1, \\ 0,&otherwise.\end{array} \right. \tag{11}\]  
Finally, the DFT of \(\hat{h}_{\text{W-DFT}}[n]\) is performed to obtain the denoised CFR. The MSE of \(k\)th subcarrier of the plus-window DFT algorithm is [11]  
\[\text{MSE}_{\text{W-DFT}}[k]=\frac{N-K}{N}\frac{\beta}{\text{SNR}}, \tag{12}\]  
where \(K\) is the number of zero-set samples. In a practical OFDM system, virtual subcarriers generally suppress out-of-band leakage. Assuming that only \(N_{\text{P}}(N_{\text{P}}<N)\) pilot subcarriers in all \(N\) subcarriers participate in channel estimation, and the set of the pilot subcarriers is \(\Omega_{\text{P}}\), then the estimated CIR under virtual subcarrier conditions can be expressed as:  
\[ \begin{array}{c}\hat{h}_{\text{V}}[n]=\frac{1}{N}\sum\limits_{k=0}^{N-1}\hat {H}_{\text{LS}}[k]R_{\text{P}}[k]e^{j\frac{2\pi}{N}kn}\\ =\sum\limits_{m=0}^{N-1}h[m]r_{\text{P}}[[n-m]]_{N}+\frac{1}{N}\sum\limits_{k=0 }^{N-1}\frac{W[k]R_{\text{P}}[k]}{X[k]}e^{j\frac{2\pi}{N}kn}\\ =h_{\text{V}}[n]+\bar{w}[n],\end{array}  \tag{13}\]where \(\left[[\cdot]\right]_{N}\) is the modulo N operation, \(h_{\text{V}}[n]\) is the channel component of the CIR, \(\hat{w}[n]\) is the noise component of the CIR with a variance of \(\tilde{\sigma}_{\text{wt}}^{2}=\frac{N_{\text{P}}}{N^{2}}\sigma_{wf}^{2}\), and \(R_{\text{P}}[k]\) and \(r_{\text{P}}[n]\) are the frequency domain window of the pilot and its time-domain response, respectively, and  
\[R_{\text{P}}[k]=\left\{ \begin{array}{ll}1,&\text{$k\in\Omega_{\text{P}}$,}\\ 0,&\text{otherwise.}\end{array} \right. \tag{14}\]  
\[r_{\text{P}}[n]=\frac{1}{N}\sum_{k=0}^{N-1}R_{\text{P}}[k]e^{\frac{j2\pi}{N}kn}. \tag{15}\]  
Appendix A shows that the MSE of \(k\)th subcarrier of the plus-window DFT algorithm in the presence of virtual subcarriers can be approximated as:  
\[\widetilde{\text{MSE}}_{\text{V-W-DFT}}[k]\approx\sum_{n=0,n\in\mathcal{O}}^ {N-1}E\Big{[}\left|h_{\text{V}}[n]\right|^{2}\Big{]}+\sum_{n=0,n\
otin \mathcal{O}}^{N-1}E\Big{[}\left|\tilde{w}[n]\right|^{2}\Big{]}, \tag{16}\]  
where \(\mathcal{O}\) is the set of samples which are set to zero in \(\hat{h}_{\text{V}}[n]\). In practical systems, it is obvious that the ideal multipath channel component \(h_{\text{V}}[n]\) cannot be obtained directly from \(\hat{h}_{\text{V}}[n]\). Fortunately, since \(E\Big{[}\left|\hat{h}_{\text{V}}[n]\right|^{2}\Big{]}=E\Big{[}\left|h_{\text{ V}}[n]\right|^{2}\Big{]}+E\Big{[}\left|\tilde{w}[n]\right|^{2}\Big{]}\), Equation (16) can be rewritten as:  
\[