## 3 Target Doppler Rate Estimation  
Typically, the signal received by the passive FSR antenna after initial processing is presented in the TF domain as a quasi-linear frequency-modulated waveform in accordance to Equation (10). Next, by using the Radon transform the Doppler rate is estimated at the crossing point which determines the kinematic parameters of the target [20; 26; 28; 29]. As shown by [PERSON] et al. [24; 25], the Hough transform can be used equivalently as a discrete realization of the Radon method, which may be used for the estimation of the frequency slope of the signal in the TF domain. This approach is very fast and can be implemented in real-life systems, however, some details have to be taken into account. Namely, classical TF representations suffer from the limited resolution resulting from the Heisenberg-Gabor uncertainty principle [30], which spoils the estimation accuracy in this case. Especially when the SNR is low or when several objects cross the baseline at the same time, the method proposed by [PERSON] et al. may require additional processing steps. Widely speaking, in many practical applications the short-time Fourier transform (STFT) based approach may be insufficient due to the finite resolution of the TF plane. Additionally, the resolution is strongly dependent on the processing parameters, such as window type, window width, overlap, etc. Even a high-SNR signal can be distributed incorrectly over the TF plane if the processing parameters are badly conditioned. One of the popular and widely applied methods for TF resolution enhancement is TF reassignment [30; 31; 32; 33]. This method can be implemented through the classical STFT-based method, as well as using a recursive version presented by [PERSON] et al. [34]. The latter is particularly interesting due to the possibility of its efficient implementation and the fast operation of the energy relocation \((t,\omega)\mapsto(\hat{t},\hat{\omega})\). As both the recursive and fast Fourier transform (FFT)-based implementations are equivalent, the FFT-based method is used in this paper as an example of the technique. In general, the STFT of the signal \(x(t)\) can be computed as follows:  
\[F_{x}^{h}(t,\omega)=\int_{\mathbb{R}}x(\tau)h^{*}(t-\tau)e^{-j\omega\tau} \mathrm{d}\tau=M_{x}^{h}(t,\omega)e^{j\phi_{x}^{h}(t,\omega)}, \tag{11}\]  
where \(j=\sqrt{-1}\), \((\cdot)^{*}\) is the complex conjugate, \(\mathbb{R}\) denotes the set of real numbers, \(M_{x}^{h}(t,\omega)\) is the amplitude, and \(\phi_{x}^{h}(t,\omega)\) is the phase of the transform. The energy distribution, commonly called a spectrogram, is defined as a squared absolute value of the STFT and is given by  
\[E_{x}^{h}(t,\omega)=|F_{x}^{h}(t,\omega)|^{2}, \tag{12}\]  
where \(|\cdot|\) denotes the absolute value operator. Using the relationship defined by [PERSON] [35], Equation (11) can be transformed into a complex phase as follows:  
\[\Phi_{x}^{h}(t,\omega)=\ln\left(F_{x}^{h}(t,\omega)\right)=\Lambda_{x}^{h}(t, \omega)+j\phi_{x}^{h}(t,\omega), \tag{13}\]  
where \(\Lambda_{x}^{h}(t,\omega)=\ln\left(M_{x}^{h}(t,\omega)\right)\). The concept of complex phase is widely used in the literature as an effective tool for the estimation of signal parameters in the TF domain, which is also applied in the considered approach. For TF reassignment, the relocation operators have to be estimated, which correspond to vectors for both the time and frequency axes along which the energy has to be moved. In the investigated approach, the reassignment operators may be estimated respectively [31]:  
\[\hat{t}(t,\omega)=-\Im\left(\frac{\partial\Phi_{x}^{h}(t,\omega)}{\partial \omega}\right)=t-\Re\left(\frac{\partial\Phi_{x}^{h}(t,\omega)}{\partial\omega }\right)=t-\Re\left(\frac{F_{x}^{Th}(t,\omega)}{F_{x}^{h}(t,\omega)}\right), \tag{14}\]  
\[\hat{\omega}(t,\omega)=\omega+\Im\left(\frac{\partial\Phi_{x}^{h}(t,\omega)}{ \partial t}\right)=\omega+\Im\left(\frac{F_{x}^{Th}(t,\omega)}{F_{x}^{h}(t, \omega)}\right), \tag{15}\]  
where \(\Re\) is the real and \(\Im\) is the imaginary part.