# A Novel GNSS Interference Detection Method Based on Smoothed Pseudo-Wigner-Hough Transform
## 3 Time-Frequency Transforms
It is also visible that the Wigner-Ville distribution exhibits improved time-frequency resolution properties over the spectrogram. From Figure 4b, it can be seen that three cross interfering terms emerge among the transient Gaussian components separated between different positions in the time or frequency domains within the time-frequency plane. It is also clear that the magnitudes corresponding to the oscillatory cross interfering terms are twice as those of the related auto-terms of the analyzed signal. Although a number of time-frequency desirable properties can be obtained with the Wigner-Ville distribution, due to its bilinear form of the time-frequency distribution, the distribution inevitably presents serious cross interfering terms in the time-frequency  
Figure 3: The spectrogram of a nonstationary signal composed of three transient Gaussian components by adopting the Hamming type of window function; (**a**) 15 samples window length; (**b**) 31 samples window length. domain. The presence of the cross-interfering terms in the time-frequency plane does not possess any physical meaning, making signal understanding and correct interpretation near impossible. Figure 4: The time-frequency distributions of a nonstationary signal composed of three transient Gaussian components. (**a**) Wigner–Ville distribution (WVD); (**b**) WVD’s contour; (**c**) Pseudo-Wigner–Ville distribution (PWVD); (**d**) PWVD’s contour; (**e**) Smoothed pseudo-Wigner–Ville distribution (SPWVD); (**f**) SPWVD’s contour. A reasonable approach to partially mitigate the influence of cross-interfering terms is to introduce a low-pass filtering window function in the Wigner-Ville distribution, resulting in the concept of pseudo-Wigner-Ville distribution [9; 11; 15; 29]. We know that the Wigner-Ville distribution approach applies equal weights temporally; therefore, if we have an interest in concentrating on the signal near a given time instant \(t\), then we can multiply the instantaneous auto-correlation function \(y_{a}\big{(}t+\frac{\tau}{2}\big{)}y_{a}^{*}\big{(}t-\frac{\tau}{2}\big{)}\) by an analysis window function \(h(\tau)\). Thus, the pseudo-Wigner-Ville distribution can be obtained. The pseudo-Wigner-Ville distribution can be explained as the common Wigner-Ville distribution with a window function in the direction of time, which can be expressed as [9; 11; 15; 29]:  
\[PWVD(t,\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h(\tau)y_{a}\Big{(}t+ \frac{\tau}{2}\Big{)}y_{a}^{*}\Big{(}t-\frac{\tau}{2}\Big{)}\mathrm{e}^{- \mathrm{i}\omega\tau}\mathrm{d}\tau \tag{11}\]  
where the window function \(h(\tau)\) can be used to control \(PWVD(t,\omega)\), which usually acts as an even function and presents its peak value around the time instant \(\tau=0\). In the pseudo-Wigner-Ville distribution method, the adoption of the analysis window function makes its distribution more local in the time-frequency domain. This is beneficial for the suppression of the cross-interfering terms for the analyzed signal with multi-components. The time-windowing operation can be considered to be equal to the frequency filtering applied in the Wigner-Ville distribution approach. In this way, the corresponding cross interfering term localized among the time-shifted signal components can be alleviated to some extent. For example, the calculated pseudo-Wigner-Ville distribution for the analyzed signal is shown in Figure 4c with the corresponding contour presented in Figure 4d. It can be seen that the cross-term between the pair of auto-terms located in different time positions is effectively mitigated. This is in contrast to the cross-term between the pair of auto-terms distributed in the same time position, which still exists in the time-frequency domain. The cross-interfering term localized among the signal components with the same time position in the time-frequency plane can be further suppressed by adopting the smoothing window function with the two-dimensional form expressed in the directions of both time and frequency.