## 4 Interference Detection Based on Joint TF Analysis by Using RSPWVD
### Reassignment Method  
Compared with WVD, SPWVD can be used to effectively depress the influence of the cross-interfering terms of a multi-component signal, but its TF concentration and localization properties decrease somewhat. In order to improve the TF aggregation properties in SPWVD, a reassignment method has been considered. From Equation (25), we can know that the two-dimensional smoothing function \(\Pi(t-s,\upomega-\uptheta)\) determines a certain TF region at the neighborhood nearby the point \((t,\upomega)\), inside which a weighted average of the WVD \(W_{y_{a}}(s,\uptheta)\) of the signal \(y_{a}(t)\) is performed. However, these mean values may not be symmetrically distributed around a certain point \((t,\upomega)\), which is the geometrical center of the TF region. Consequently, the point \((t,\upomega)\) is not truly representative for such a region. In contrast, the energy gravity center of such a region is more approximate to represent the local energy distribution of the analyzed signal. The local energy distribution \(\Pi(t-s,\upomega-\uptheta)C(s,\uptheta;\Pi)\) of [PERSON]'s class TF distribution can be assumed as the distribution of mass, and it is better to assign the total mass to the gravity center rather than to its geometrical center. In this paper, the reassignment method is considered to relocate each value of [PERSON]'s class distribution \(C(t,\upomega)\) at any point \((t,\upomega)\) to another point \((\hat{t},\hat{\upomega})\), which is the gravity center of the signal's energy distribution around the point \((t,\upomega)\). Then, the reassigned [PERSON]'s class TF distribution can be defined as follows:  
\[C^{(r)}(t^{\prime},\upomega^{\prime};\Pi)=\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}C(t,\upomega;\Pi)\updelta(t^{\prime}-\hat{t}(y_{a};t,\upomega ))\updelta(\upomega^{\prime}-\hat{\upomega}(y_{a};t,\upomega))\mathrm{d}t \mathrm{d}\upomega \tag{29}\]  
where:  
\[\hat{t}(y_{a};t,\upomega)=\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}s\Pi(t-s,\upomega-\uptheta)W_{y_{a}}(s,\uptheta)\mathrm{d}s\mathrm{d }\uptheta}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Pi(t-s,\upomega- \uptheta)W_{y_{a}}(s,\uptheta)\mathrm{d}s\mathrm{d}\uptheta} \tag{30}\]  
\[\hat{\upomega}(y_{a};t,\(C^{(r)}(t^{\prime},\mathbf{\omega}^{\prime};\Pi)\) is [PERSON]'s class TF distribution after reassignment. Therefore, if a suitable smoothing kernel is selected, all of the bilinear TF distributions after reassignment are able to mitigate the cross-terms and keep high TF concentration properties at the same time. In particular, when the reassignment method is applied to SPWVD, RSPWVD can be obtained, which is calculated as follows:  
\[RSPWVD(t^{\prime},\mathbf{\omega}^{\prime};g,h)=\int_{-\infty}^{ \infty}\int_{-\infty}^{\infty}SPWVD(t,\mathbf{\omega};g,h)\delta(t^{\prime}-\hat{t }(y_{a};t,\mathbf{\omega}))\cdot \tag{32}\] \[\delta(\mathbf{\omega}^{\prime}-\hat{\mathbf{\omega}}(y_{a};t,\mathbf{\omega} ))\text{d}t\text{d}\mathbf{\omega}\]  
where:  
\[\hat{t}(y_{a};t,\mathbf{\omega})=t-\frac{SPWVD(t,\mathbf{\omega};\uptau_{g},h)}{2\uppi SPWVD (t,\mathbf{\omega};g,h)} \tag{33}\]  
\[\hat{\mathbf{\omega}}(y_{a};t,\mathbf{\omega})=\mathbf{\omega}+j\frac{SPWVD(t,\mathbf{\omega} ;g,D_{h})}{2\uppi SPWVD(t,\mathbf{\omega};g,h)} \tag{34}\]  
with \(\uptau_{g}=tg(t)\) and \(D_{h}(t)=\frac{\text{d}h(t)}{\text{d}t}\).