# A Haar Wavelet Decision Feedback Channel Estimation Method in OFDM Systems
## 5 Simulation Results
### The Performance in Static Channel  
The performance metrics BER and NMSE under static CDT1 channel are shown in Figures 6 and 7, respectively. The performance metrics BER and NMSE under static CDT6 channel are shown in Figures 8 and 9, respectively.  
Figure 5: The threshold \(\hat{\sigma}\) under CDT1 and CDT6 channels.  
Figure 6: Bit error rate (BER) performance of quadrature phase shift keying (QPSK) modulated CP-OFDM and wavelet-OFDM systems under CDT1 channel.  
As can be seen from Figure 6, the BER performance of above several channel estimation methods all decreases with the increase of SNR. Under the same conditions, the DF and LS channel estimation methods have higher BER results among the four channel estimation methods. Although LS channel estimation method has about 0.3 dB gains compared with DF method at BER of \(10^{-2}\), the DF method has high data transmission rate and spectrum efficiency. The proposed Haar wavelet method has the lowest BER among the four channel estimation methods. At the BER of \(10^{-3}\), the proposed Haar wavelet method can provide about 0.5 dB and 1.5 dB SNR gains compared with DFT and DF methods, respectively. Hence, under static slight frequency-selectivity CDT1 channel conditions, the Haar wavelet method can be used to estimate the channel properties.  
As can be seen from Figure 7, as the SNR increases in the CP-OFDM and wavelet-OFDM systems, the simulation results of four channel estimation methods are decreased in terms of NMSE. Under the same conditions, the proposed Haar wavelet method has the lowest NMSE performance. At the NMSE of \(10^{-1}\), compared with the DFT channel estimation method, the proposed Haar wavelet method has about 7.5 dB SNR gains.  
As can be seen from Figures 8 and 9, the BER and NMSE performance are presented in CDT6 channel. The CDT6 channel has stronger frequency-selectivity than CDT1 channel. Hence, the performance under CDT6 channel is worse than CDT1 channel. With the increase of SNR, the simulation results show a trend of overall decreasing. Although the CDT6 channel has worse propagate condition than CDT1 channel, the proposed Haar wavelet method also has the best performance among the four channel estimation methods. In Figure 8, the simulation results about the proposed Haar wavelet method is very close to the ideal channel estimation. At the BER of \(10^{-2}\), compared with the DFT, LS, and DF channel estimation methods, the proposed Haar wavelet method has about 0.9 dB, 2.5 dB, and 3 dB BER gains, respectively. In Figure 9, compared with the DFT, LS, and DF methods, the proposed Haar wavelet method has about 8.4 dB, 10.3 dB, and 13.1 dB NMSE gains at the NMSE of \(10^{-1}\). Hence, the proposed Haar wavelet method has the best performance among the DF, LS, and DFT channel estimation methods.  
Figure 7: Normalized mean square error (NMSE) performance of QPSK modulated CP-OFDM and wavelet-OFDM systems under CDT1 channel.  
Figure 8: BER performance of QPSK modulated CP-OFDM and wavelet-OFDM systems under CDT6 channel.  
Figure 9: NMSE performance of QPSK modulated CP-OFDM and wavelet-OFDM systems under CDT6 channel.