# A Novel GNSS Interference Detection Method Based on Smoothed Pseudo-Wigner-Hough Transform
## 4 Smoothed Pseudo-Wigner-Hough Transform
When considering the GNSS signal \(y(t)\), the Wigner-Hough transform can generally be explained as the mapping from the time-frequency domain to the parameter domain \((f,k)\), which can be realized by the integral transform, expressed as [16; 25]:  
\[WHT(f,k)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}y_{a}\Big{(}t+\frac{ \tau}{2}\Big{)}y_{a}^{*}\Big{(}t-\frac{\tau}{2}\Big{)}\mathrm{e}^{-\mathrm{j}2 \pi(f+kt)\tau}\mathrm{d}\tau\mathrm{d}t \tag{14}\]  
where \(y_{a}(t)\) represents the analytic form of the signal \(y(t)\) to be analyzed, and \(f\) and \(k\) are the initial frequency and chirp rate for the analyzed signal, respectively. By using the form of polar coordinates, Wigner-Hough transform in Equation (14) can be rewritten as [25]:  
\[WHT(\rho,\theta)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}y_{a}\Big{(}t +\frac{\tau}{2}\Big{)}y_{a}^{*}\Big{(}t-\frac{\tau}{2}\Big{)}\mathrm{e}^{- \mathrm{j}2\pi\frac{1}{\mathrm{i}\mathrm{i}\mathrm{i}\mathrm{i}}(\rho-\cos \theta\cdot t)\tau}\mathrm{d}\tau\mathrm{d}t \tag{15}\]  
where \(\theta\) ranges in the interval \([0,\pi)\). The Wigner-Hough transform may also be explained as a line integral of Wigner-Ville distribution, expressed as [25]:  
\[ \begin{array}{ll}WHT(f,k)&=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} WWD(t,v)\delta(v-f-kt)\mathrm{d}v\mathrm{d}t\\ &=\int_{-\infty}^{+\infty}WVD(t,f+kt)\mathrm{d}t\\ &=\int_{-\infty}^{+\infty}WVD(t,\frac{1}{\mathrm{i}\mathrm{i}\mathrm{i}\mathrm{i }\mathrm{i}}(\rho-\cos\theta\cdot t))\mathrm{d}t\end{array}  \tag{16}\]  
where \(\delta(\cdot)\) is the Dirac delta function. The pseudo-Wigner-Ville distribution has been introduced to partially suppress the cross-interfering terms with the Wigner-Ville distribution [9; 15], as can been verified in Section 3; the Hough transform can be further combined with the pseudo-Wigner-Ville distribution to develop the pseudo-Wigner-Hough transform to improve GNSS interference detection performance [25]. The definition of pseudo-Wigner-Hough transform is given as:  
\[PWHT(f,k)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}h(\tau)y_{a}\Big{(} t+\frac{\tau}{2}\Big{)}y_{a}^{*}\Big{(}t-\frac{\tau}{2}\Big{)}\mathrm{e}^{- \mathrm{j}2\pi(f+kt)\tau}\mathrm{d}\tau\mathrm{d}t \tag{17}\]  
Similar to the Wigner-Hough transform, the pseudo-Wigner-Hough transform in Equation (17) can also be explained as the line integral of pseudo-Wigner-Ville distribution, expressed as:  
\[ \begin{array}{ll}PWHT(f,k)&=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} PWVD(t,v)\delta(v-f-kt)\mathrm{d}v\mathrm{d}t\\ &=\int_{-\infty}^{+\infty}PWVD(t,f+kt)\mathrm{d}t\\ &=\int_{-\infty}^{+\infty}PWVD(t,\frac{1}{\mathrm{i}\mathrm{i}\mathrm{i} \theta}(\rho-\cos\theta\cdot t))\mathrm{d}t\end{array}  \tag{18}\]  
From Section 3, it is shown that smoothed pseudo-Wigner-Ville distribution can be used to mitigate the cross-interfering terms with the bilinear time-frequency distribution. However, this is achieved at the cost of a deteriorating time-frequency localization property [9]. Therefore, in this paper, the smoothed pseudo-Wigner-Ville distribution can be further combined with the Hough transform to propose a novel smoothed pseudo-Wigner-Ville distribution-Hough transform (i.e., smoothed pseudo-Wigner-Hough transform, SP-WHT) in interference detection for GNSS receivers. The smoothed pseudo-Wigner-Hough transform can be then defined as:\[SPWHT(f,k)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+ \infty}g(u)h(\tau)y_{a}\Big{(}t+u+\frac{\tau}{2}\Big{)}y_{a}^{*}\Big{(}t+u- \frac{\tau}{2}\Big{)}\mathrm{e}^{-\mathrm{i}2\pi(f+kt)\tau}\mathrm{d}\tau \mathrm{d}u\mathrm{d}t \tag{19}\]  
Moreover, the \(\rho-\theta\) transform space should be discrete for the Hough transform based GNSS interference detection methods.