## 4 Interference Detection Based on the Reassigned Spectrogram
### Reassignment Method  
The most simple cross-term-free distribution of the Cohen's class is the spectrogram, which can be expressed in Equation (29) using the smoothing kernel \(\Pi(t,\omega)=W_{h}(t,\omega)\), where \(W_{h}(t,\omega)\) denotes the WVD of the window function \(h(t)\). This smoothing operation removes the cross-terms present in the WVD \(W_{y_{a}}(t,\omega)\) of the analyzed signal \(y_{a}(t)\), but it also smears out the signal auto-terms. The mitigation of the cross-terms can be posed as a kernel function \(\Pi(t,\omega)\) design problem subject to the constraint of preserving the good TF resolution of \(C(t,\omega;\Pi)\). In order to improve the TF aggregation properties of \(C(t,\omega;\Pi)\), the existing solution consists of a non-uniform post-processing of TF distributions guided by local signal characteristics, which can be implemented by the reassignment method. The principle of the reassignment method is to compensate the faults in mapping the TF distribution \(C(t,\omega;\Pi)\) by relocating the value of the neighboring TF energy at its current location \((t,\omega)\) to the gravity center \((\hat{t},\hat{\omega})\) rather than its geometrical center. The starting point of the reassignment method is Equation (29). From Equation (29), we can know that the two-dimensional smoothing function \(\Pi(t-s,\omega-\uptheta)\) determines a certain TF region at the neighborhood nearby the point \((t-s,\omega-\uptheta)\), inside which a weighted average of the WVD \(W_{y_{a}}(s,\uptheta)\) of the signal \(y_{a}(t)\) is performed. [PERSON]'s class \(C(t,\omega;\Pi)\) is then the average of the TF energy located in a domain centered on \((t,\omega)\) and delimited by the essential support of \(\Pi\). However, this mean value may not be symmetrically distributed around this geometrical center \((t,\omega)\); consequently, the geometrical center \((t,\omega)\) is not truly representative of such a TF region. In contrast, the TF energy gravity center of such a TF region is more approximate to represent the local energy distribution of the analyzed signal. The local TF energy distribution \(\Pi(t-s,\omega-\uptheta)W_{y_{a}}(s,\uptheta)\) of [PERSON]'s class can be assumed as a distribution of mass, and it is better to assign the mass to the gravity center. The TF energy averaging in Equation (29) leads to the attenuation of the oscillating cross-terms, but it also presents the undesirable effect of broadening the signal components in the TF plane. Therefore, the solution to avoid this effect is the adoption of the reassignment method, which can be generalized to all bilinear TF distributions. In order to improve the TF concentration of the signal components, the reassignment method is considered to relocate each value of [PERSON]'s class \(C(t,\omega;\Pi)\) at any point \((t,\omega)\) to another point \((\hat{t},\hat{\omega})\), which is the gravity center of the signal's TF energy distribution around the point \((t,\omega)\). According to Equation (29), the spectrogram in Equation (19) can be also expressed as a two-dimensional smoothing of the WVD, which can be obtained by using the smoothing kernel \(\Pi(t,\mathbf{\omega})=W_{h}(t,\mathbf{\omega})\), written as follows:  
\[S(t,\mathbf{\omega})=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_ {h}(t-s,\mathbf{\omega}-\mathbf{\theta})W_{y_{a}}(s,\mathbf{\theta})\text{d}s\text{d}\theta \tag{31}\]  
This above expression shows that the value of the spectrogram \(S(t,\mathbf{\omega})\) at any point \((t,\mathbf{\omega})\) is a weighted sum of WVD \(W_{y_{a}}(s,\mathbf{\theta})\) at the neighborhood nearby the point \((t-s,\mathbf{\omega}-\mathbf{\theta})\); thus, \(S(t,\mathbf{\omega})\) is the sum of a whole TF energy distribution located around its geometrical center \((t,\mathbf{\omega})\).