## 4 Interference Detection Based on the Reassigned Spectrogram
### Reassignment Method
In order to avoid the resulting signal component broadening, an effective solution is to move this average from the point \((t,\mathbf{\omega})\), where \(S(t,\mathbf{\omega})\) is computed for the center of gravity \((\hat{t},\hat{\mathbf{\omega}})\) of its composing energy distributions. When the reassignment method is applied to the spectrogram in Equation (31), the obtained reassigned spectrogram can be written as follows:  
\[S^{(r)}(t^{\prime},\mathbf{\omega}^{\prime})=\frac{1}{2\pi}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}S(t,\mathbf{\omega})\delta(t^{\prime}-\hat{t}(y_{a};t,\mathbf{ \omega}))\delta(\mathbf{\omega}^{\prime}-\hat{\mathbf{\omega}}(y_{a};t,\mathbf{\omega})) \text{d}t\text{d}\mathbf{\omega} \tag{32}\]  
where:  
\[\hat{t}(y_{a};t,\mathbf{\omega})=\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}s\Pi(t-s,\mathbf{\omega}-\mathbf{\theta})W_{y_{a}}(s,\mathbf{\theta})\text{d}s \text{d}\mathbf{\theta}}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Pi(t-s, \mathbf{\omega}-\mathbf{\theta})W_{y_{a}}(s,\mathbf{\theta})\text{d}s\text{d}\mathbf{\theta}} \tag{33}\]  
\[\hat{\mathbf{\omega}}(y_{a};t,\mathbf{\omega})=\frac{\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}\theta\Pi(t-s,\mathbf{\omega}-\mathbf{\theta})W_{y_{a}}(s,\mathbf{\theta })\text{d}s\text{d}\mathbf{\theta}}{\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\Pi(t-s,\mathbf{\omega}-\mathbf{\theta})W_{y_{a}}(s,\mathbf{\theta})\text{d}s \text{d}\mathbf{\theta}} \tag{34}\]  
where \(S^{(r)}(t^{\prime},\mathbf{\omega}^{\prime})\) is the spectrogram after reassignment; and \(\delta(t)\) denotes the Dirac impulse. The reassignment operation in Equation (32) leads to the construction of a modified version of the spectrogram, whose value at any point \((t^{\prime},\mathbf{\omega}^{\prime})\) is the sum of all of the TF energy distributions moved to this point. The most interesting property of the reassigned spectrogram is considered that the respective time and frequency reassigned location values \(\hat{t}(y_{a};t,\mathbf{\omega})\) and \(\hat{\mathbf{\omega}}(y_{a};t,\mathbf{\omega})\) can also be derived from the partial derivatives of the phase of the STFT of \(y_{a}(t)\), written as follows:  
\[\hat{t}(y_{a};t,\mathbf{\omega})=-\frac{\partial\Phi_{y_{a}}(t,\mathbf{\omega})}{ \partial\mathbf{\omega}} \tag{35}\]  
\[\hat{\mathbf{\omega}}(y_{a};t,\mathbf{\omega})=\mathbf{\omega}+\frac{\partial\Phi_{y_{a}} (t,\mathbf{\omega})}{\partial t} \tag{36}\]  
where \(\Phi_{y_{a}}(t,\mathbf{\omega})\) is the phase of the STFT of the analyzed signal \(y_{a}(t)\), _i.e._, \(\Phi_{y_{a}}(t,\mathbf{\omega})=\text{arg}(STFT_{y_{a}}(t,\mathbf{\omega}))\). The above-mentioned expressions in Equations (35) and (36) can be interpreted respectively as the local group delay and the local instantaneous frequency of the signal \(y_{a}(t)\) observed in the TF domain imposed by the analysis window \(h(t)\). Although the expressions in Equations (35) and (36) are physically meaningful, unfortunately, they do not lead to an efficient implementation of the reassignment operation for discrete time signals, since the derivatives should be approximated by first order differences. Thus, the calculation method for the reassignment operators remains unused despite its attractive theoretical background.