## 4 Statistical Quality Assessment of GNSS Signal
### Frequency Domain Features  
By finding the spectral content of the signal using the fast Fourier transform (FFT), the following attributes were calculated:  
1. Normalized Spectrum Bandwidth (NSBW): \[NSBW=\frac{f_{max}-f_{min}}{f_{c}},\] where \(f_{c}\) is the centre frequency. 2. Normalized Spectrum Kurtosis (NSK): \[NSK=\frac{\sum_{k=f_{1}}^{f_{2}}\left(f_{k}-S_{c}\right)^{4}P_{k}}{\sum_{k=f_{1 }}^{f_{2}}p_{k}s^{4}},\]where \(S_{c}\) is the spectral centroid. It is given by  
\[S_{c}=\frac{\sum_{i=1}^{m}f_{k}m_{i}}{\sum_{i=1}^{m}m_{i}}\]  
where \(f_{1}\) and \(f_{2}\) are the bounds of the frequency band, \(f_{k}\) is the frequency, and \(P_{k}\) is the normalized power and m-magnitude of the bin number. 3. Normalized Spectrum Flatness (NSF):  
\[NSF=\frac{\exp\left(\frac{1}{N}\sum_{k=1}^{N}ln(X(k))\right)}{\frac{1}{N}\sum_ {k=1}^{N}(X(k))},\]  
where \(X(k)\) is the magnitude of the k-th frequency bin in the spectrum. N is the total number of frequency bins. 4. Ratio of the Variance to the Squared Mean of the Normalized Spectrum (RVSM):  
\[\text{RVSM}=\frac{\sigma(X)}{\left(\sum_{k=1}^{N}X(k)\right)^{2}},\]  
where X is the normalized spectrum, \(\sigma(X)\) is the variance of the normalized spectrum, and \(N\) is the total number of elements in the normalized spectrum. 5. Single-Frequency Energy Aggregation (SFEA):  
The summation of the squares of the Fourier transform coefficients results in an energy aggregation given by  
\[E_{total}=\sum_{i=1}^{n}\lvert X_{i}\rvert^{2}\]  
6. Ratio of the Maximum Peak to the Second Maximum Peak of the Normalised Spectrum (RMPS):  
First, find the index in the spectrum \(X_{max}\) of the maximum peak in \(X\): \(i_{max}=argmax(X)\) and remove the maximum peak from \(X\), denoted as \(X^{\prime}\): \(X^{\prime}=X/\left\{X\lvert i_{max}\right\}\). Then, the second peak is calculated as \(max(X^{\prime})\). When computing the feature correlation using the Pearson correlation coefficient, the closer the value is to +1, the better we can deduce from the plot that the feature being considered is the feature most correlated with the target variable (response). The correlation coefficient is defined as  
\[r(a,b)=\frac{\sum_{i=1}^{n}\left(a_{i}-\overset{\sim}{a}\right)\left(b_{i}- \overset{\sim}{b}\right)}{\sqrt{\sum_{i=1}^{n}\left(a_{i-\overset{\sim}{a} \right)^{2}}}\sqrt{\sum_{i=1}^{n}\left(b_{i-\overset{\sim}{b}}\right)^{2}}}, \tag{5}\]  
where the two attribute samples are denoted by \(a_{i}\) and \(b_{i}\), and the mean values of the samples are given as \(\overset{\sim}{a}\) and \(\overset{\sim}{b}\). The correlation coefficient represents the degree of positive or negative correlation between two variables, with its value ranging from \(-1.0\) to +1.0. the darker colours signify weaker correlations, while the lighter shades denote stronger correlations. Figure 3a,b show the correlation maps of the weakly and strongly correlated features of the deteriorated GNSS signal in both domains. Figure 3: _Cont._  
Figure 4a illustrates the power levels of different types of interference versus six frequency domain features. Notably, as the power level increases from 40 to 60 dB, there is a sudden rise in the normalized spectral kurtosis value. For chirp-type interference, the normalized spectral flatness, RVSM, and RMPS values are slightly elevated at mid-range interference power levels. Figure 4: _Cont._  
Figure 3: (**a**) Correlation maps for time domain features (**i**) Clean GNSS (**ii**) Chirp (**iii**) Pulse (**iv**) CWI (**v**) Spoofing (**vi**) Multipath (**vii**) MCWI. (**b**) Correlation maps for frequency domain features (**i**) Clean GNSS (**ii**) Chirp (**iii**) Pulse (**iv**) CWI (**v**) Spoofing (**vi**) Multipath (**vii**) MCWI. \[NSK=\frac{\sum_{k=f_{1}}^{f_{2}}\left(f_{k}-S_{c}\right)^{4}P_{k}}{\sum_{k=f_{1 }}^{f_{2}}p_{k}s^{4}},\]0 _Cont_. In contrast, the frequency domain feature, depicted in Figure 4b, shows a spurious peak in the mean value for pulse-type interference around a 30 dB interference power level.