# An Anti-Interference Method Based on Energy Residual Searching in GNSS Positioning Applications
## 5 Simulation and Results

Table 1: Experimental parameters in interference detection test. ference signals limits its application. This trade-off is common in time-frequency analysis and requires careful selection and optimization based on specific application requirements. Through the analysis of this phenomenon, we have gained further insight into the limitations of time-frequency analysis tools when dealing with complex interference signals. In Figure 3, it can be seen that the LFM signal exhibited an energy concentration characteristic at the optimal fractional order. By simply applying spectral line removal at this order, the LFM interference could be filtered out, thus avoiding the issues associated with the trade-off between time and frequency resolution. The squared magnitude \(|Y_{p}(u)|^{2}\) of the FrFT of the interference-affected GNSS signal is clearly depicted on the \(u\)-\(p\) plane. A distinct and sharp energy peak was observed, indicating the presence of frequency-sweep interference within the received GNSS signal. The energy peak was evidently concentrated in a very limited region on the \(u\)-\(p\) plane. In contrast, the energy contributions from known useful GNSS signals and white noise were distributed more diffusely across the \(u\)-\(p\) plane. For any fractional order of the FrFT, this portion of energy remained ineffective in accumulation and was negligible in intensity on the \(u\)-\(p\) plane.As illustrated in Figure 4, the LFM interference signals demonstrated a significant energy concentration effect at specific fractional orders, where the signal  
Figure 3: The 3D representation of the FrFT for the LFM signal. Figure 2: Using the Hamming window function for spectrogram analysis of GNSS signals: (**a**) shows the result with a Hamming window length of 63, while (**b**) shows the result with a Hamming window length of 31.  
energy converged into a sharp pulse. After spectral line removal, the energy of the LFM interference signal can be significantly reduced, as shown in Figure 4b. In contrast, the GNSS signals and noise signals at the same fractional order showed a more dispersed energy distribution without a clear concentration trend. This phenomenon indicates that using the fractional Fourier transform to separate LFM interference at specific fractional orders is effective. It can be observed that, prior to filtering, the receiver was unable to capture a distinct correlation peak, as shown in Figure 5a. However, after filtering, the receiver was able to successfully capture the correlation peak, as illustrated in Figure 5b. Therefore, it is evident that this method demonstrates effective filtering performance for LFM interference. Linear frequency modulation (LFM) interference has a severe impact on GNSS receiver performance, leading to a significant degradation in positioning accuracy and, in extreme cases, a complete failure of positioning functionality. Specifically, when the energy of the LFM interference signal is relatively weak, it can prevent the receiver from reliably acquiring a sufficient number of satellite signals, resulting in inaccurate positioning solutions, as illustrated in Figure 6a. The acquisition and the tracking of satellite signals are adversely affected by the interference, leading to noticeable scattering and decreased accuracy in positioning results. When the energy of the LFM interference signal is strong, the GNSS receiver may be unable to capture valid satellite signals, thereby preventing the computation of positioning solutions, as shown in Figure 5b. Furthermore, interfering signals may increase the carrier-to-noise ratio within the channel, further impacting positioning accuracy. The interference mitigation method based on energy residual searching proposed  
Figure 4: Comparison of before and after spectral line removal: (**a**) before spectral line removal; (**b**) after spectral line removal. Figure 5: Comparison of the searching space before and after interference mitigation: (**a**) search result before processing; (**b**) search result after processing.