# An Anti-Interference Method Based on Energy Residual Searching in GNSS Positioning Applications
## 4 Anti-Interference Method Based on Energy Residual Searching  
### Constructing the Energy Residual Function  
First, a predefined FrFT is applied to the digitized downconverted analytic signal using a preset fractional order (although this order may not be the optimal one). This step aims to concentrate the energy of the LFM interference signal in the transform domain, without causing the energy of the noise and GNSS signals to concentrate. The fractional Fourier transform of the analytic signal can be expressed as \[ \begin{split} F^{p}[x](\frac{m}{2\triangle x})=\frac{A_{a}}{2 \triangle x}exp[j\pi(\gamma-\beta)(\frac{m}{2\triangle x})^{2}].\\ \sum_{n=-(\triangle x)^{2}}^{(\triangle x)^{2}}exp[j\pi\beta( \frac{m-n}{2\triangle x})^{2}]exp[j\pi(\gamma-\beta)(\frac{m}{2\triangle x})^{2 }]x(\frac{n}{2\triangle x})\end{split}  \tag{3}\]  
where \(F\) represents the Fourier transform operator; \(p\) denotes the fractional order of the Fourier transform; \(j\) is the imaginary unit; \(A_{a}\) denotes the transformation coefficient; \(\alpha\) is the transformation angle; \(\beta=csc\alpha\); \(\gamma=cot\alpha\); \(m\) denotes a discrete sampling point variable in the transformation domain; \(\triangle x=\sqrt{T_{obs}/T_{s}}\) denotes the normalized width; and \(T_{obs}\) denotes the time duration of the processed signal. For simplicity, we denote \(F^{p}[x](\frac{m}{\triangle x})\) as \(x_{p}[\frac{m}{\triangle x}]\). The summation part in the above equation is a discrete convolution, which can be computed using FFT, with a complexity of \(NlogN\). To determine the threshold value for spectral line excision, the N-sigma method is an appropriate choice. The rationale behind this is that when the LFM signal is transformed into a pulse function in the fractional domain, its energy is concentrated on a few spectral lines, while the remaining data points exhibit relatively low values and a more uniform distribution. The N-sigma method effectively identifies these significant spectral lines, thereby enabling the determination of a reasonable threshold value. Specifically, the threshold can be expressed as  
\[Th=K*\delta+\mu \tag{4}\]  
where \(\mu\) represents the mean value of the spectral line amplitudes; \(\delta\) denotes the standard deviation of the spectral line amplitudes; and \(K\) denotes the selected multiplication factor used to control the stringency of identifying outlier spectral lines. After calculating the threshold value, spectral components in the transform domain with amplitudes exceeding this threshold are excised. This step effectively suppresses the high-amplitude components of the signal in the transform domain. Specifically, the process of spectral line excision can be represented as  
\[x_{p}[\frac{m}{\triangle x}]=\left\{ \begin{array}{ll}0,&x_{p}[\frac{m}{ \triangle x}]>Th\\ x_{p}[\frac{m}{\triangle x}],&x_{p}[\frac{m}{\triangle x}]\leq Th\end{array} \right. \tag{5}\]  
Based on the computed threshold value, spectral components with amplitudes exceeding this threshold are removed. The energy of the transformed-domain signal is then calculated after the spectral line removal:  
\[E_{1}=\sum_{n=-(\triangle x)^{2}/2}^{(\triangle x)^{2}/2}|x[\frac{n}{\triangle x }]|^{2} \tag{6}\]  
We calculate the signal energy in the transformed domain before the spectral line removal. According to [PERSON]'s theorem, the energy of the transformed-domain signal before removal is equal to the energy of the analytic signal:  
\[E_{2}(p)=\sum_{m=-(\triangle x)^{2}/2}^{(\triangle x)^{2}/2}[|x_{p}|\frac{m}{ \triangle x}|^{2}] \tag{7}\]  
Based on the aforementioned, the energy residual function is constructed. The energy residual refers to the reduction in signal energy after the spectral line removal, reflecting the extent to which high-amplitude components have been removed. Specifically, it can be expressed as  
\[res(p)=|E_{1}-E_{2}(p)^{2}| \tag{8}\]The significance of the energy residual lies in its ability to quantitatively assess the suppression effect of the LFM interference signal in the transform domain.