## 3 Simulation  
The simulation experiment procedure is shown in Figure 3. A sinusoidal signal (amplitude 1, frequency is \(f_{r}=f_{0}+f_{d}\), \(f_{0}=40\) MHz to avoid negative frequencies, signal doppler frequency shift \(f_{\rm d}=\pm\) 4 MHz, phase randomly varying in \((-\pi,\pi)\)) is generated to simulate Doppler shift and the incoherent accumulation in real systems. A 200 ns wide, 20 kHz repetition frequency pulsed signal, with Gaussian distributed trailing on both sides, is modulated to mimic the atmospheric wind field's backward Mie scattered echo. After adding Gaussian white noise with a set standard deviation to the simulated echo, RW, classic windows (Blackman and five-term MSD), and different-order HSCW are added in the time domain. The zero-padding FFT algorithm obtains the spectrum, which is then incoherently summed over different periods. Finally, the energy center method is used as a spectral correction algorithm to estimate the frequency of the simulated echo. Figure 2: Comparison of HSCWs of different orders. At a sampling frequency of 250 MHz, the simulation system mimics the echo signal pulse width with about 50 effective sampling points. To narrow the window function's main lobe while ensuring the coherent Doppler wind lidar's spatial resolution, the system takes every 64 points as a range gate. This is equivalent to using the rectangular window function to truncate time domain signals every 64 points. After zero padding the analog echo signals intercepted by the window function to 128 and 256 points, the results of the FFT algorithm are shown in Figure 4a,b, in which the horizontal coordinates represent the frequency points, the frequency value represented by each unit frequency point is \(\Delta f=f_{s}/K\) (\(f_{s}\) is the system sampling rate, and \(K\) is the number of FFT points), and the vertical coordinates represent the signal amplitude. The black line represents the RW intercepted signal spectrum, and the red line represents the first-order HSCW intercepted signal spectrum. It can be seen that there is a serious spectral leakage of the RW intercepted signal spectrum. The spectral lines of the HSCW intercepted signal are concentrated in the main lobe when compared with the RW interception. However, due to the loss of energy during the interception, the amplitude of the spectral lines is lower. This phenomenon does not lead to the reduction in the system SNR in the case of the system where the signal is superimposed on the noise. A comparison between Figure 4a,b shows that, in the case where the HSCW intercepts a 64-point signal echo, zero padding the echo data to 128 points and 256 points can improve the spectral resolution without affecting the width of the main lobe of the window function. This results in an increase in the number of spectral lines within the main lobe, which hinders the subsequent spectral correction. To simulate the real-world conditions where wind field echoes are weaker than noise, Gaussian white noise (std = 5) is added to the simulated echo in the range gate. A 128-point FFT is used to obtain the spectrum. As shown in Figure 5a, strong noise prevents the direct extraction of Doppler shift. To improve the SNR, spectral information from different cycles is incoherently accumulated. Figure 5b-f show the results for different accumulation times (N = 250, 500, 1000, 2000, 5000). The horizontal axis represents the frequency points (1.95 MHz each), and the vertical axis is the normalized amplitude. The black, red, blue, brown, purple, and green lines represent the RW, first-order HSCW, second-order HSCW, Blackman, five-term MSD, and third-order HSCW results. For easier analysis, the accumulation results of Gaussian white noise are subtracted after simulating the echoes, and the results of each window function are normalized by dividing by the amplitude of the highest spectral line. Figure 3: Simulation experiment flowchart. Figure 6 shows the gain process and SNR at different incoherent accumulation times for five window functions.