# Joint Channel Estimation Algorithm Based on DFT and DWT
## 3 DFT-Based Channel Estimation Algorithm
### Channel Estimation Algorithm Based on Segmented Threshold DFT
Therefore, the decision threshold of the effective taps in the proposed channel estimation algorithm based on segment threshold DFT can be expressed as:  
\[T=\left\{ \begin{array}{ll}P_{\text{max}},&\tilde{P}<2\tilde{\sigma}_{wt}^{2 },\\ 2\tilde{\sigma}_{wt}^{2},&\text{otherwise},\end{array} \right. \tag{21}\]  
and the specific calculation of \(\tilde{\sigma}_{wt}^{2}\) is described in detail in Section 5. The specific steps used in the ST-DFT channel estimation algorithm are as follows:  
Step 1: Based on the received signal and training sequence, estimate the frequency domain response of the channel \(\hat{P}_{\text{LS}}\) estimated using the LS algorithm;  
Step 2: Perform \(N\)-point IFFT to \(\hat{P}_{\text{LS}}\) to get the time-domain channel impulse response \(\hat{h}_{\text{V}}\) with power dispersion;  
Step 3: Calculate the threshold \(T\) according to Equation (21);  
Step 4: Perform a threshold determination on \(\hat{h}_{\text{V}}\), and get \(\hat{h}_{\text{ST-DFT}}\). \[\hat{h}_{\text{ST-DFT}}[n]=\left\{ \begin{array}{ll}\hat{h}_{\text{V}}[n],& \left|\hat{h}_{\text{V}}[n]\right|^{2}\geq T,\\ 0,&\left|\hat{h}_{\text{V}}[n]\right|^{2}<T;\end{array} \right. \tag{22}\]  
Step 5: Perform \(N\)-point FFT to \(\hat{h}_{\text{ST-DFT}}\) to obtain the channel frequency response estimate after denoising \(\hat{H}_{\text{ST-DFT}}\). It is assumed that the set of samples set to zero in \(\hat{h}_{\text{V}}[n]\) is \(\Theta\) and the number of samples is \(K\). The ST-DFT algorithm yields different \(\Theta\) for each estimation of different \(\hat{h}_{\text{V}}[n]\). For the convenience of the engineering implementation, the expected values \(E\left[\left|\hat{h}_{\text{V}}[n]\right|^{2}\right]\) and \(E\left[\left|\tilde{w}[n]\right|^{2}\right]\) in Equation (17) are replaced by the instantaneous value in the single estimated CIR. The MSE of \(k\)th subcarrier of the ST-DFT channel estimation algorithm can be approximated as:  
\[\widetilde{\text{MSE}}_{\text{ST-DFT}}[k]\approx\sum\limits_{n=0,n\in\Theta }^{N-1}\left|\hat{h}_{\text{V}}[n]\right|^{2}+\sum\limits_{n=0}^{N-1}\left| \tilde{w}[n]\right|^{2}-2\sum\limits_{n=0,n\in\Theta}^{N-1}\left|\tilde{w}[n ]\right|^{2}. \tag{23}\]  
For a single estimated CIR (\(\hat{h}_{\text{V}}[n]\)) and a given effective tap decision threshold \(T\), among all samples with the same or similar channel component power values, those samples that are always superimposed with less power noise are more likely to be classified as noisy samples and are set to zero. That is, the average power of the noise component superimposed on those zero-set samples is no greater than that of the noise component superimposed on all samples in the estimated CIR. At the same time, the average power of the superimposed noise component on taps (such as \(K\) taps) that are set to zero should not be less than the average power of the \(K\) noise component with the smallest superimposed power value on all samples in the CIR. Therefore, as shown in Appendix B, the range of values for the total noise power \(\sum\limits_{n=0,n\in\Theta}^{N-1}\left|\tilde{w}[n]\right|^{2}\) contained in all zero-set samples can be expressed as:\[\left(\frac{K}{N}+\left(1-\frac{K}{N}\right)\ln\left(1-\frac{K}{N}\right)\right) \frac{N_{\text{P}}}{N}\sigma_{s\text{-}wf}^{2}\leq\sum\limits_{n=0,n\in\Theta}^ {N-1}\left|\bar{w}[n]\right|^{2}\leq\frac{KNp}{N^{2}}\sigma_{s\text{-}wf}^{2}, \tag{24}\]  
where \(\sum\limits_{n=0}^{N-1}\left|\bar{w}[n]\right|^{2}=\frac{N_{\text{P}}}{N} \sigma_{s\text{-}wf}^{2}\), \(\sigma_{s\text{-}wf}^{2}\) is the frequency domain noise variance on the active subcarriers in the current symbol.