# A Haar Wavelet Decision Feedback Channel Estimation Method in OFDM Systems
## 4 Proposed Haar Wavelet Decision Feedback Channel Estimation Method  
To have high spectrum and data transmission efficiency like the DF channel estimation method, at the same time, the AWGN in time-domain can be effectively removed. This paper proposes a Haar wavelet decision feedback channel estimation method. The threshold obtained by the Haar wavelet decomposition can be utilized to suppress the AWGN in time-domain effectively. Compared with the time-domain threshold channel estimation method in [16] which also uses Haar wavelet, this paper uses DF channel estimation method to obtain the initial channel estimation coefficients and improves the spectrum and data transmission rate. Paper [16] uses pilot-based LS method to obtain the initial channel estimation coefficients which decreases the spectrum and data transmission efficiency. In the wavelet-OFDM system proposed in this paper, IFFT and FFT operations are used to combine different subcarriers into a composite time-domain signal. But papers [33; 34; 35] use inverse discrete wavelet transform (IDWT) and DWT instead of IFFT and FFT, which aim to suppress the ICI and ISI. Paper [35] also investigated different wavelets to adapt to the specific channel environment. The Haar wavelet is used as subcarriers in [33; 34; 35]. In this paper, Haar wavelet decomposition is used for suppressing noise existing at the estimated channel coefficients in time-domain. In the practical application, the received signals are disturbed by the AWGN. After the wavelet decomposition, signals are divided into the important and unimportant coefficients [16]. The important coefficients contain the useful information about the signal and the AWGN is concentrated on the unimportant coefficients. The important coefficients also called approximation coefficients can be obtained based on low-pass filter. The unimportant coefficients also called detail coefficients can be obtained based on high-pass filter. The Haar wavelet has the advantage of simple and easy implementation. In addition, the channels used in this paper are multipath fading channels which have the characteristic of frequency selectivity. Haar wavelet is more suitable for this channel characteristic [36]. Hence, this paper uses Haar wavelet to decompose the channel response. According to Haar wavelet, the frequency response of low-pass filter \(F(\mathrm{e}^{\mathrm{i}\omega})\) and high-pass filter \(G(\mathrm{e}^{\mathrm{i}\omega})\) can be respectively expressed as [16]  
\[F(\mathrm{e}^{\mathrm{i}\omega})=\mathrm{e}^{\mathrm{i}\omega/2}\cos(\frac{ \omega}{2}), \tag{7}\]  
\[G(\mathrm{e}^{\mathrm{i}\omega})=-\mathrm{j}\mathrm{e}^{\mathrm{i}\omega/2} \sin(\frac{\omega}{2}), \tag{8}\]  
where \(\omega\) is radian frequency. The detail coefficients are obtained by high-pass filter and the frequency of noise is relatively higher than that of the signal. As a result, noise existing at high frequency is the dominant components in the detail coefficients. The standard deviation of the detail coefficients corresponding to the noise added to the signal can be obtained by [16]  
\[\hat{\sigma}=\frac{\mathrm{median}(|\boldsymbol{d}_{i}|)}{0.3045}, \tag{9}\]  
where \(\boldsymbol{d}_{i}\) are the detail coefficients obtained by the Haar wavelet decomposition. \(\mathrm{median}(\cdot)\) is the operation of obtaining the mid-value of a vector. In [16], constant value 0.6745 is used as the denominator in Equation (9). The channel environment used in this paper is different from the paper [16]. Simulation verifies that 0.3045 is an optimal value to obtain good system performance. The procedure of the proposed Haar wavelet method is presented in Figure 4.  
As shown in Figure 4, to obtain high data transmission rate and spectrum efficiency, the initial CFR \(\hat{\mathbf{H}}_{\text{DF}}\) is obtained by the DF channel estimation method. Then, the time-domain CIR \(\hat{\mathbf{h}}_{\text{DF}}\) is obtained by the IFFT operation.