## 4 Interference Detection Based on Joint TF Analysis by Using RSPWVD
### Spwvd  
The unsatisfactory results obtained with the existing TF distributions justify the search for better tools; one way of achieving this is to start from the general form of quadratic representations. All of these existing TF distributions could be written in a generalized form, which can be used to facilitate the design of desirable TF transforms. This class of transform is known as [PERSON]'s class Cohen (1964), and the definition of [PERSON]'s class of bilinear (or quadratic) TF distributions can be written as follows:  
\[\begin{split} C(t,\upomega)&=\frac{1}{4\pi^{2}} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}AF(\uptau,\uptheta)g(\uptau, \uptheta)e^{-j(\uptheta+\upomega\uptau)}\uptheta\uptheta\uptau\\ &=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\int_{-\infty}^{\infty}y_{a}(s+\frac{\uptau}{2})y_{a}^{*}(s-\frac{ \uptau}{2})g(\uptau,\uptheta)e^{-j[\uptheta(t-s)+\upomega\uptau]}\uptheta \uptheta\uptau\end{split} \tag{24}\]  
where \(AF(\uptau,\uptheta)\) is the ambiguity function defined in Equation (17); and \(g(\uptau,\uptheta)\) is a two-dimensional parametrization function defined in the ambiguity function domain, which is called the kernel function of [PERSON]'s class. This kernel function determines the properties of the bilinear TF distribution, which is often a low-passing function and normally serves to mask out the interference in the original Wigner-Ville representation. When considering \(g(\uptau,\uptheta)=1\), WVD can be obtained.  
[PERSON]'s class can be also rewritten as the double convolution of the WVD of the signal \(y_{a}(t)\) and a two-dimensional smoothing function, provided as follows:  
\[\begin{split} C(t,\upomega;\Pi)&=W_{y_{a}}(t, \upomega)**\Pi(t,\upomega)\\ &=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\Pi(t-s,\upomega-\uptheta)W_{y_{a}}(s,\uptheta)\uptheta\uptheta\end{split} \tag{25}\]  
\[\Pi(t,\upomega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(\uptau,\uptheta )e^{-j(\uptheta\uptau+\uptheta\uptau)}\uptheta\uptheta \tag{26}\]  
where (\(**\)) denotes the double convolution operation, \(\Pi(t,\upomega)\) is a two-dimensional smoothing function and \(W_{y_{a}}(s,\uptheta)\) is the WVD of the signal \(y_{a}(t)\). Different TFRs can be obtained from the fundamental WVD by applying a different smoothing function \(\Pi(t,\upomega)\). When considering WVD, \(\Pi(t,\upomega)=\updelta(t)\updelta(\upomega)\).  
If a separable smoothing function is considered, it can be written as the product of windows from both time and frequency domains:  
\[\Pi(t,\upomega)=g(t)H(-\upomega) \tag{27}\]  
where \(H(\upomega)\) is the Fourier transform of the window function \(h(t)\), which allows the smoothing of the cross-interfering terms oscillating in parallel with the frequency axis (frequency smoothing); and the window function \(g(t)\) allows the smoothing of the cross-interfering terms oscillating in parallel with the temporal axis (temporal smoothing). Obviously, the wider the spreading, the more the smoothing. Therefore, the smoothing in the ambiguity function domain combined with the parametrization function allows both the suppression of the cross-terms and the preservation of the auto-ambiguity terms of the analyzed signal.  
Since the cross-terms with the WVD are strongly oscillating, the most effective way of removing cross-term interference is to apply two-dimensional low-pass filtering in the ambiguity domain. The resulting two-dimensional convolution of the WVD in Equation (25) defines the smoothed version of PWVD, that is SPWVD, which can be written as follows:  
\[SPWVD(t,\upomega;g,h)=\int_{-\infty}^{\infty}h(\uptau)\int_{-\infty}^{\infty} g(t^{\prime}-t)y_{a}(t^{\prime}+\frac{\uptau}{2})y_{a}^{*}(t^{\prime}-\frac{ \uptau}{2})e^{-j\upomega\uptau}\mathrm{d}t^{\prime}\mathrm{d}\uptau \tag{28}\]  
Therefore, an independent and progressive control can be applied to the WVD in both time and frequency directions. The independency of \(h(t)\) and \(g(t)\) makes SPWVD more flexible to reduce the cross-terms present in WVD.