## 4 Results
(**d**) Hough transform of RSPWVD, JNR is 6 dB. (**e**) Hough transform of WVD, JNR is 12 dB. (**f**) Hough transform of RSPWVD, JNR is 12 dB. In order to assess the performance between the Hough transform of WVD and RSPWVD, the root mean square errors (RMSE) of the rate estimation and initial frequency estimation for chirp interference are used and displayed as a function of JNR in Figure 6. The setting parameters of the experiments are the same as in Figure 4. Figure 6a shows that the RMSE of the chirp one rate estimated by the Hough transform of WVD is the same as that estimated by the Hough transform of RSPWVD when JNR ranges from 0 dB to 12 dB. Similarly, the RMSE of the chirp two rate estimated by the Hough transform of WVD is the same as that estimated by the Hough transform of RSPWVD when JNR ranges from 0 dB to 12 dB. In Figure 6a, it is obvious that the Hough transform of RSPWVD as well as that of WVD provides the same RMSE of the rate estimation for each chirp when JNR ranges from 0 dB to 12 dB. In Figure 6b, the RMSE of the initial frequency for chirp one estimated by the Hough transform of RSPWVD is close to \(10^{-2}\) when JNR is equal to 0 dB and 2 dB, while the RMSE of the initial frequency for chirp one estimated by the Hough transform of WVD is about \(0.5\times 10^{-2}\) when JNR is equal to 0 dB and 2 dB. However, when JNR equals or surpasses 4 dB, the Hough transform of RSPWVD provides the same RMSE of the initial frequency for chirp one. On the other hand, the RMSE of the initial frequency for chirp two estimated by the Hough transform of WVD is close to \(0.4\times 10^{-2}\) when JNR ranges from 0 dB to 12 dB. The RMSE of the initial frequency for chirp two estimated by the Hough transform of RSPWVD is close to \(0.1\times 10^{-2}\) when JNR ranges from 0 dB to 12 dB. Figure 6b depicts that the Hough transform of RSPWVD offers a poorer RMSE of the initial frequency estimation for chirp one compared with the Hough transform of WVD when JNR is below 4 dB and provides the same RMSE of the initial frequency estimation as that of WVD for chirp one when JNR equals or surpasses 4 dB. For chirp two, the proposed method provides a better RMSE of the initial frequency estimation compared with the Hough transform of WVD. In addition, another experiment is performed by the proposed method under the condition where one chirp is adopted and its instantaneous frequency ranges from 30 MHz to 50 MHz. The sample length is 512 and the sweep period is 2.56 us, 5.12 us and 10.24 us. The results of RMSE are shown in Table 2. In Table 2, the estimation of initial frequency is close to \(0.5\times 10^{-2}\) and the estimation of the chirp rate is about \(0.6\times 10^{-2}\) when the sweep period is 2.56 us. However, when the sweep period increases to 5.12 us, the estimation of the initial frequency decreases by approximately \(1.5\times 10^{-2}\), and the estimation of chirp rate decreases by almost \(0.7\times 10^{-2}\). Finally, when the sweep period reaches 10.24 us, the estimation of initial frequency continues to decline by about \(1.7\times 10^{-2}\) and the estimation  
Figure 6: RMSE of the chirp rate and initial frequency estimation for chirp interferences present in BDS B1I signals by Hough transform combined by WVD and RSPWVD versus JNR. The \(C_{0}/N\) is 43 dB-Hz. (**a**) RMSE of chirp rate. (**b**) RMSE of initial frequency. of the chirp rate reduces to a level near \(0.8\times 10^{-2}\). From Table 2, it is obvious that the estimation accuracy of the initial frequency as well as the chirp rate declines as the sweep period increases. The computational requirements of the Hough transform of WVD and RSPWVD are shown in Table 3 [18]. \(N\) is sample length, \(M\) is the length of the time window and \(L\) is the length of the frequency window. The computational requirements of the Hough transform of WVD consist of the computational requirements of the Hough transform as well as those of WVD.