# Joint Channel Estimation Algorithm Based on DFT and DWT
## 4 DWT-Based Channel Estimation Algorithm  
The wavelet transform is often used to analyze signals with different frequencies and time resolution and to decompose the signal into detail coefficients and approximate coefficients, which are defined as high-frequency coefficients and low-frequency coefficients, respectively. Compared with the traditional Fourier transform, the wavelet transform has advantages in terms of accurately decomposing and reconstructing finite, aperiodic, and non-stationary signals. For the channel frequency response \(\hat{H}_{\text{LS}}\) obtained by the LS channel estimation, the Mallat algorithm is used for discrete wavelet transform, and the output wavelet coefficients can be expressed as [16]:  
\[ \begin{array}{c}C_{j,k}=\sum\limits_{n}C_{j-1,n}h^{*}(n-2k),\\ D_{j,k}=\sum\limits_{n}D_{j-1,n}g^{*}(n-2k),\end{array}  \tag{27}\]  
where \(C_{j,k}\) is the approximate wavelet coefficient, \(D_{j,k}\) is the detailed wavelet coefficient, \(j\) is the order of wavelet decomposition, \(h^{*}\) and \(g^{*}\) are a set of orthogonal mirror filters, and \(\hat{H}_{\text{LS}}=C_{0}=D_{0}\). In the practical wavelet decomposition, the length of the coefficients after each layer of wavelet decomposition is not the number of coefficients in the previous layer halved because of the addition of filters. The number of wavelet coefficients after decomposition in each layer can be expressed as:  
\[L_{j}=floor\left(\frac{L_{j-1}-1}{2}\right)+N_{\text{FB}}, \tag{28}\]  
where \(L_{j}\) represents the wavelet coefficients after the \(j\)th wavelet decomposition, \(floor(\cdot)\) denotes the downward rounding, and \(N_{\text{FB}}\) is the length of the filter. The total wavelet coefficients \(L_{\text{total}}\) is the total number of wavelet coefficients in the multi-layer decomposition. At present, there is no unified standard for the selection of wavelet bases, and all walks of life choose according to their actual needs [19; 20]. Since this paper studies the channel estimation algorithm for the block pilot, the Db4 wavelet with better continuity is selected as the mother wavelet [16]. Having too few wavelet decomposition layers will lead to an unsatisfactory denoising effect, and having too many decomposition layersmay misjudge the effective signal as noise and remove the valid signal. At present, the wavelet-denoising algorithm in the field of channel estimation is generally decomposed into 1\(\sim\)3 layers [16; 17; 18]. This paper does not discuss the optimal number of decomposition layers for the DWT. To ensure that the effective signal is preserved as much as possible under high SNR condition, the intermediate value of two layers is selected as the wavelet decomposition layers. The discrete wavelet transform was performed on the received signal interfered with by Gaussian white noise. Since Gaussian white noise is independently distributed, the white noise after wavelet transform is also highly random in the wavelet domain. Therefore, the detailed coefficient obtained by the DWT corresponds to Gaussian white noise, and its size and variance will change greatly with the transformation of the wavelet scale. On the contrary, the approximate coefficient corresponds to the useful signal, and its size and variance change little with the transformation of the wavelet scale [21]. The noise mixed in the signal can be filtered out by thresholding the detail coefficients. In this paper, the unbiased likelihood estimation was used as the decision threshold for wavelet-denoising with the following calculation expression [22]:  
\[T_{\text{DWT}}=\frac{median(|D_{i}|)}{0.6745}, \tag{29}\]  
where \(D_{i}\) is the \(i\)th wavelet coefficient in the detailed coefficients decomposed by Equation (27), \(median(\cdot)\) means to take the middle value. In order to give the processed wavelet coefficients have better continuity and smoothness, a soft threshold function is used for processing, which is given by [16]:  
\[D^{\prime}_{j,k}=\left\{