# S-Band Doppler Wave Radar System
## 3 MORSE Radar System
### Algorithm Process  
After the down-conversion performed by the RF analog front end and digital receiver, the echoes are transformed into baseband signals. The sample rate of the baseband signals is 125 KHz, and every single baseband signal has 512 sample points. To obtain the Doppler spectra, two FFT will be applied to one Doppler dataset consisting of 128 echoes. After a 512-point FFT applied to every echo, the range spectra, which is named FT1 data, are obtained. The frequency resolution is approximately 244 Hz because the sample rate is 125 KHz. Now we have 128 range spectra and every range spectrum has 512 points. Because every point in a range spectrum represents a range cell. The FT1 data can be viewed from another aspect, which is to say that there are 512 range cells and every range cell has 128 sample points. The sample rate here is the Doppler sample rate. So the Doppler spectra, which is named FT2 data, are then calculated by applying another 128-point FFT to the 128-point data of every range cell. The example of FT1 data and FT2 data is showed in Figure 4. The calculation process of FT2 data is illustrated in Figure 5. The period for the frequency sweep is 4112 us, which means that the sample rate of the Doppler spectra is 243.19 Hz. Hence, the frequency resolution of the Doppler spectra is 1.899 Hz. For one range cell, total time consumption of an 128-point Doppler spectrum sample is 128 \(\times\) 4112 us \(\approx\) 0.526 s.  
Figure 4: Example of FT1 data and FT2 data. The data are applied 3-point smooth. (**a**) FT1 data of 56 th echo. (**b**) FT2 data of 56 th range cell (420 m far from MORSE). Figure 3: Radar antennas array. The central frequency shift of the Doppler spectra is fundamental to the velocity series. In the MORSE system, the steps to extract central frequency shift are as follows:  
1. Calculate the peak SNR of a Doppler power density spectrum. First, we take 15 points on the left edge and 15 points on the right edge of a Doppler power density spectrum. Then, we calculate the average of the two sets of points and consider the lower value as the power of noise \(P_{Noise}\). Furthermore, we choose the largest value in the spectrum as the maximum signal power \(P_{Signal}\). Eventually the SNR is calculated with the following equation: \(SNR=10\lg\frac{P_{Signal}}{P_{Noise}}\)
2. Find the bandwidth. We search for 3 points consecutive with the SNR value lower than the threshold (normally 10 dB) from the location of the maximum value to the left and right. Then the locations of the two set of 3 points are treated as the left and right boundaries of the bandwidth are \(B_{L}\) and \(B_{R}\)
3. Extract the Doppler frequency shift. We use the spectrum moment method to calculate the frequency shift as follows: \(\Delta f=\frac{\int_{B_{L}}^{B_{R}}f\times P(f)df}{\int_{B_{L}}^{B_{R}}P(f)df}\)  
The doppler velocity can then be calculated with the Doppler effect after determining the central Doppler frequency shift of the Doppler spectra. Because the time consumption of an 128-point Doppler spectrum sample is 0.526 s, the doppler velocity of one range cell is calculated every 0.526 s. With each antenna, we calculate 256 doppler velocity to generate one doppler velocity series for every range cell, which takes approximately 134.626 s. The example of Doppler frequency shift spectra and Doppler velocity spectra is showed in Figure 6. Then according to the Formula (6) and (7), the wave height spectrum of that range cell is calculated. In the end, with the spectrum moment method, the wave parameters are obtained using the following equations [28]. Figure 5: The calculation process of FT2 data.