## 5 Performance Evaluation
In practice, the spectrogram approach cannot be used to provide the instantaneous frequency estimate for the interfering signal, although sometimes, a good approximation can be achieved. In Figure 9a, the WVD of the GPS L1 signal with the sweep interference is depicted, and correspondingly, the contour of the computed WVD is presented in Figure 9b. The interfering term presents a linear behavior in frequency, which is well localized in the restricted portion of the TF plane. The price that is paid for high TF resolution with the WVD approach is the undesirable cross-terms. Due to the interaction of the different signal components, the presence of the serious cross-terms is apparently observed in Figure 9b, which makes interpretation quite difficult and brings much error for the estimation of the instantaneous frequency of the interfering signal. In Figure 10a, the PWVD of the GPS L1 signal with the sweep interference is depicted, and correspondingly, the contour of the computed PWVD is presented in Figure 10b. It is clear that although the sweep interfering term presents a linear behavior in the TF plane, the PWVD presents less accurate TF localization precision, since the TF concentration property of the PWVD is attenuated. In addition, the cross-terms can be still observed in the TF plane by using the PWVD approach. Figure 8: Spectrogram of the GPS L1 signal with sweep interference. The spectrogram has been evaluated by using a Hamming window. \(C/N_{0}\) = 46 dB-Hz, JNR = \(-\)1 dB. In order to mitigate the cross-interfering terms present in the WVD, a two-dimensional smoothing function is used to smooth out the interfering signal in both the time and frequency domains. In Figure 10(a), the SPWVD of the interfered GPS L1 signal is provided, where the peaks of the SPWVD are clearly observed, denoting the sweep interference contribution. The corresponding contour of the SPWVD is provided in Figure 10(b), and it is easy to find that the undesired sweep interference shows a linear behavior in frequency, which is localized in the linear portion of the TF plane. From Figure 10(b), it is very clear that the undesired cross-interfering terms are partially mitigated by the adoption of the  
Figure 10: Pseudo Wigner–Ville distribution of the GPS L1 signal with sweep interference. \(C/N_{0}=\) 46 dB-Hz, JNR \(=-1\) dB. Figure 9: Wigner–Ville distribution of the GPS L1 signal with sweep interference. \(C/N_{0}\) = 46 dB-Hz, JNR \(=-1\) dB. two-dimensional smoothing function in the SPWVD. Although the adoption of the smoothing window in SPWVD is beneficial to suppress the cross-terms in the TF plane, it smears localized components, leading to less accurate TF localization of the auto-terms of the signal, as compared to the WVD approach. The disadvantage of this filtering window operation is that it limits the original excellent TF resolution features. Therefore, in order to remove the cross-terms present in the quadratic TF distribution and to preserve the good TF energy concentration properties at the same time, an improved TF distribution by adopting RSPWVD has been proposed in interference detection for GNSS receivers. This proposed joint TF distribution of the interfered GPS L1 signal is depicted in Figure 12a, and the corresponding contour of the calculated TF distribution of RSPWVD is provided in Figure 12b. From the results, it is clear that the undesired sweep interfering term presents a very strict linear behavior in the TF plane with the proposed joint TF distribution based on RSPWVD, which provides representations that are easy to interpret in interference detection for GNSS receivers. This proposed improved TF analysis technique effectively eliminates the cross-term artifacts present in the quadratic TF distributions, which shows good readability in the TF plane. In the reassigned method, the smoothing window is moved from the geometrical center to the energy gravity center of the TF distribution.