## 3 Target Doppler Rate Estimation
\(\hat{t}(t,\omega)\) denotes the relocation along the \(t\)-axis and \(\hat{\omega}(t,\omega)\) expresses the reassignment vector along the frequency axis. In contrast to another energy concentration technique known from the literature, for example TF synchrosqueezing transform [32], the reassignment method allows strong concentration to be obtained, however, this technique is irreversible. In Equation (14), the expression \(\mathcal{T}h=th(t)\) is a window multiplied by the linear time ramp with a root in 0, and \(\mathcal{D}h\) in Equation (15) denotes the first order derivative of the analyzing window \(\mathcal{D}h=\frac{\mathrm{d}h(t)}{\mathrm{d}t}\). In fact, these operations can be interpreted as follows:  
\[\partial F_{x}^{h}(t,\omega)\Big{/}\partial t=\int x(\tau)\frac{\partial h^{* }(t-\tau)e^{-j\omega(t-\tau)}}{\partial t}\mathrm{d}\tau=F_{x}^{Th}(t,\omega), \tag{16}\]  
as well as:  
\[\partial F_{x}^{h}(t,\omega)\Big{/}\partial\omega=\int x(t-\tau)h^{*}(\tau) \frac{\partial e^{-j\omega\tau}}{\partial\omega}\mathrm{d}\tau=-\int x(t- \tau)h^{*}(\tau)j\tau e^{-j\omega\tau}\mathrm{d}\tau=-jF_{x}^{Th}(t,\omega), \tag{17}\]  
which leads to:  
\[\frac{\partial\ln(F_{x}^{h}(t,\omega))}{\partial t}=\frac{\partial F_{x}^{h} (t,\omega)}{\partial t}\frac{1}{F_{x}^{h}(t,\omega)}=\frac{F_{x}^{Th}(t, \omega)}{F_{x}^{h}(t,\omega)} \tag{18}\]  
and  
\[\frac{\partial\ln(F_{x}^{h}(t,\omega))}{\partial\omega}=\frac{\partial F_{x}^{ h}(t,\omega)}{\partial\omega}\frac{1}{F_{x}^{h}(t,\omega)}=-j\frac{F_{x}^{Th}(t, \omega)}{F_{x}^{h}(t,\omega)} \tag{19}\]  
which can be directly applied in Equations (14) and (15). This means that the reassignment operators can be easily computed through the STFT method using the modified analyzing window, which increases the utility of the method and reduces the computational effort. Equivalently, the method may be implemented using a recursive filter bank, as described in [34]. Finally, the energy relocation using the reassignment method can be expressed as [31]  
\[R_{x}^{h}(t,\omega)=\iint_{\mathbb{R}^{2}}|F_{x}^{h}(t,\omega)|^{2}\delta(t- \hat{t}(t,\omega))\delta(\omega-\hat{\omega}(t,\omega))\mathrm{d}t\mathrm{d}\omega, \tag{20}\]  
where \(\delta(\cdot)\) denotes the Dirac distribution. The distribution given by Equation (20) results in strongly concentrated energy on the TF plane, with an enhanced readability and separated components. In fact, the reassignment method usually does not relocate the maximum of the energy but only attracts the surrounding distribution, hence the signal localization remains stable whilst the readability of the transform is improved. The TF reassignment is a widely used technique in many applications, e.g., ultrasound signal processing [36], audio signal analysis [37; 38], or sonar applications [39]. However, despite the high potential of this method, it is still not very popular in the radar community. Namely, in the literature one can find results for the energy concentration in micro-Doppler signature analysis [40; 41; 42], characterization of frequency shift keying (FSK) signals [43], improving the quality of inverse synthetic aperture radar (ISAR) imaging [44] as well as in direction of arrival estimation [45] and Doppler radar tomography imaging [46]. The novelty presented in this paper is to apply the TF reassignment method and combine it with the Hough transform that aims to extract the Doppler rate of the target with the improved accuracy comparing the classical method existing in the literature. The valuable properties of the TF reassignment method prompted the authors to apply this technique to analyze radar signals in a passive FSR application. The \"energy gathering\" properties applied in such an application may improve the accuracy of the Doppler rate estimation in passive FSR systems. This, in fact is the novelty proposed in this paper, since to the authors knowledge there is no similar application that uses the TF reassignment method to improve the accuracy of the Doppler rate estimate in passive FSR system.