# A Novel GNSS Interference Detection Method Based on Smoothed Pseudo-Wigner-Hough Transform
## 3 Time-Frequency Transforms  
The common and simple technique used for evaluating the time-varying signal frequency characteristic is short-time Fourier transform (STFT) [15; 29]. In the simple case for the continuous-time nonstationary signal, short-time Fourier transform can be represented as [10; 15; 29]:  
\[STFT(t,\omega)=\int_{-\infty}^{+\infty}y_{a}(\tau)h(\tau-t)\mathrm{e}^{-\mathrm{ j}\omega t}\mathrm{d}\tau \tag{8}\]  
where \(h(t)\) is the analysis window function, which is typically real and even and centered around zero, \(y_{a}(t)\) represents the analytic signal to be analyzed, and \(STFT(t,\omega)\) is a linear function depending on the selected window function \(h\). The spectrogram can be defined as the squared magnitude of short-time Fourier transform, expressed as [10; 15; 29]:  
\[SPEC(t,\omega)=\left|STFT(t,\omega)\right|^{2} \tag{9}\]  
The nonstationary signal, which consists of three transient Gaussian components with different positions in time and frequency, is shown in Figure 1. By changing the types of analysis window functions, the spectrogram of the signal is shown in Figure 2. In Figure 2a, the Chebyshev window is selected, and in Figure 2b, the Kaiser window is chosen, and the window length is set to 15 samples in both cases. From Figure 2, it can be observed that the spectrogram shows different time-frequency resolutions based on the selected type of window function, which usually provides insufficient properties of the time-frequency resolution. Figure 1: The nonstationary signal composed of three transient Gaussian components. Figure 2: The spectrogram of nonstationary signal composed of three transient Gaussian components; (**a**) Chebyshev window; (**b**) Kaiser window. Furthermore, in order to determine the time-frequency resolution characteristics of the spectrogram by varying the window function size in the time-frequency analyses, the spectrogram of the aforementioned signal is shown in Figure 3, where the Hamming type of the window function is adopted. In Figure 3a, the window length is set to 15 samples. It can be seen that the small size of the analysis window function in the time domain provides impoverished resolution in the frequency domain. From Figure 3b, when the analysis window length is set to 31 samples, it is shown that increasing the window size results in an increased resolution in the frequency domain. In this way, the limitation from the uncertainty principle with the spectrogram can be further verified, pointing to the trade-off problem of time-frequency resolution due to the adoption of the analysis window function in the spectrogram. The commonly used solution for the spectrogram's time-frequency resolution trade-off problem is to adopt Wigner-Ville distribution in the time-frequency analysis, which can be explained as [8; 9; 10; 11; 15; 29]:  
\[WVD(t,\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}R(t,\tau)\mathrm{e}^{- \mathrm{i}\omega\tau}\mathrm{d}\tau=\frac{1}{2\pi}\int_{-\infty}^{+\infty}y_{a }(t+\frac{\tau}{2})y_{a}^{*}\Big{(}t-\frac{\tau}{2}\Big{)}\mathrm{e}^{-\mathrm{ i}\omega\tau}\mathrm{d}\tau \tag{10}\]  
where \(\tau\) is the lag variable, \(R(t,\tau)=y_{a}\big{(}t+\frac{\tau}{2}\big{)}y_{a}^{*}\big{(}t-\frac{\tau}{2} \big{)}\) is the instantaneous auto-correlation function, and \((*)\) is the operation of complex conjugate. The Wigner-Ville distribution presents several desirable properties and provides a near-optimal time-frequency resolution among all the time-frequency distributions. However, its main drawback comes from undesirable cross-term interference [6; 8; 9; 10; 11; 15; 29]. An example of the Wigner-Ville distribution of the nonstationary signal is shown in Figure 4a, and the corresponding contour of the calculated Wigner-Ville distribution is shown in Figure 4b. It can be seen from Figure 4a that there are three time-frequency energy peaks, which represent the corresponding auto-terms of the analyzed signal in the time-frequency domain.