# Joint Channel Estimation Algorithm Based on DFT and DWT
## 3 DFT-Based Channel Estimation Algorithm
### Channel Estimation Algorithm Based on Segmented Threshold DFT  
To deal with the problem that the performance of the existing unified threshold DFT algorithm needs to improve under medium and low SNR conditions, this paper proposes a channel estimation algorithm based on segmented threshold DFT. By setting different thresholds in different SNR segments, the channel multipath can be estimated effectively. According to the multipath distribution characteristics in the estimated CIR, under medium and low SNR conditions, the power of the channel component in and around the channel delay spread region is relatively high, while the power of the noise component in the rest of the region is relatively high. When virtual subcarriers are present, the channel delay spread depends on the time-domain response of the pilot window function. Taking the Wi-Fi6 channel estimation scenario as an example, let \(N=256\), \(N_{\text{P}}=242\) in the HE-LFT field. The time-domainimpulse response of the pilot window function expressed by Equation (15) is shown in Figure 1.  
As seen in Figure 1, for a single-tap channel, when there is power spreading, the power is mainly concentrated in the range with a small offset. For example, for a Wi-Fi6 system with a CP length of \(1/16\) (i.e., \(N_{\text{CP}}=16\)), from Equation (15) we know that 99% of the channel power is included in \([0:N_{\text{CP}}-1,N-N_{\text{CP}}:N-1]\), which can be regarded as the channel observation area. For multipath channels, since the power of each path generally decreases with an increase in its time delay, the channel power that spreads outside the channel observation area is very limited, so the main power of the channel is still concentrated in the channel observation area. Correspondingly, the area outside the channel observation area can be regarded as the noise observation area \(\Omega_{w}\), and its range is \([N_{\text{CP}}:N-N_{\text{CP}}-1]\). Let the average power of the samples in \(\Omega_{w}\) be:  
\[\bar{P}=\frac{\sum\left|h_{\text{V}}[n]\right|^{2}}{N-2N_{\text{CP}}},n\in \Omega_{w}. \tag{18}\]  
In addition, through Equation (13), we know that the average power of the channel component and noise component in \(\Omega_{w}\) can be expressed as:  
\[\bar{P}_{u}=\frac{\sum|h_{\text{V}}[n]|^{2}}{N-2N_{\text{CP}}},\bar{P}_{w}= \frac{\sum|\tilde{w}[n]|^{2}}{N-2N_{\text{CP}}},n\in\Omega_{w}. \tag{19}\]  
Since the channel component and the noise component are independent of each other, the expected values of the sample power, the channel component power, and the mean value of the noise component power satisfy the relationship \(E[\bar{P}]=E[\bar{P}_{u}]+E[\bar{P}_{w}]\), where \(E[\bar{P}_{w}]=\tilde{\sigma}_{wt}^{2}\). The results can be divided into two cases for discussion. The first case is when \(E[\bar{P}_{u}]<E[\bar{P}_{w}]\), i.e., \(E[\bar{P}]<2E[\bar{P}_{w}]\), and the SNR is relatively low. The power of the noise component for all samples in the \(\Omega_{w}\) region can be considered to be generally greater than the power of the channel component. Under this condition, the decision threshold of the CIR effective path can be set to the maximum value of the power of the samples in the \(\Omega_{w}\) region, and all the samples in the \(\Omega_{w}\) region will be set to zero. The maximum power can be expressed as:  
Figure 1: Power of the time-domain response of the pilot window function \(|r_{\text{P}}[n]|^{2}\). \[P_{\text{max}}=\text{max}\left\{\left|\hat{h}[n]\right|^{2}\right\},n\in\Omega_{w}. \tag{20}\]  
The other case is when \(E[\hat{P}_{u}]\geq E[\hat{P}_{w}]\), i.e., \(E[\hat{P}]\geq 2E[\hat{P}_{w}]\). The SNR is relatively high. The power of the noise component for all samples in the \(\Omega_{w}\) region can be considered to be less than that of the channel component. Under this condition, the decision threshold of the CIR effective path can be set to \(2E[\hat{P}_{w}]\), in order to retain the samples with greater channel power in the \(\Omega_{w}\) region.