## 2 Analytical Description of the Proposed Algorithm  
In this section, the theory of the proposed algorithm is introduced. A block diagram of the proposed module is shown in Figure 2 and is used to illustrate the main stages of our algorithm. Figure 2: A detailed diagram of the proposed module. First, we generate a complex baseband bi-phase signal and denote it by \(X(t)\). The bi-phase signal is then sent through the traditional modules of a radar transmitter. At the receiver side and after demodulation, the return signal has the following form:  
\[R_{1}(t)=X(t-\tau)e^{-j2\pi f_{d}(t-\tau)}+n(t), \tag{1}\]  
where \(\tau\) is the target delay, \(f_{d}\) is the target Doppler effect, \(n(t)\) is the noise, and \(X(t-\tau)\) is the delayed bi-phase waveform. As a pre-processing step, the proposed module uses median filtering, a nonlinear digital filtering technique, to reduce noise and improve the performance of the subsequent processing. This filter is used as it effectively preserves edges during the denoising process [18; 19]. With an appropriate window size, the effect of this filter on the signal and noise can be assumed to have the following form:  
\[R_{2}(t)=a_{1}X(t-\tau)e^{-j2\pi f_{d}(t-\tau)}+a_{2}n(t), \tag{2}\]  
where \(a_{1}\) and \(a_{2}\) are real numbers with \(a_{1}\gg a_{2}\). The filtered signal is then subjected to a nonlinear transformation \(\mathcal{F}\), which enables us to eliminate the effect of the bi-phase and isolate the Doppler effect in the time domain. Although nonlinear transformations were also used in [20], it was primarily to compensate for very low-frequency offsets. In our case, we apply it for the blind estimation of larger Doppler shifts, starting with a coarse estimate, which is then refined in subsequent iterations. Given that we use a bi-phase waveform, we choose \(\mathcal{F}\) to be an even power function, in particular, a quadratic one, which we apply in all subsequent iterations of the algorithm. To illustrate the effect of the transformation, we choose \(\mathcal{F}\) to be a quadratic function and apply it to \(R_{2}(t)\), as shown below. \[\mathcal{F}\big{(}R_{2}(t)\big{)} =R_{2}(t)^{2}\] \[=a_{1}^{2}e^{-j4\pi f_{d}(t-\tau)}+a_{2}^{2}n^{2}(t)+2a_{1}a_{2}X( t-\tau)n(t)e^{-2\pi jf_{d}(t-\tau)}\] \[=a_{1}^{2}\big{(}e^{-j4\pi f_{d}(t-\tau)}+\epsilon^{2}n^{2}(t)+2 \epsilon X(t-\tau)n(t)e^{-2\pi jf_{d}(t-\tau)}\big{)}, \tag{3}\]  
where \(\epsilon=\frac{a_{2}}{a_{1}}\). From this equation, we can see that the nonlinear transformation \(\mathcal{F}\) serves two objectives: (1) it cancels the bi-phase waveform from the signal, as the term \(X(t-\tau)\) represents the bi-phase waveform, which oscillates between \(+1\) and \(-1\) depending on its phase. When we apply the nonlinear transformation, the term \(X(t-\tau)^{2}\) always results in \(1\), cancelling the bi-phase signal, and (2) it turns the multiplicative nature of the Doppler effect into an additive form. In other words, at the receiver, the received signal consists of the delayed signal multiplied by the Doppler effect, in addition to the noise. After transformation, the signal can be represented by three terms, as Equation (3) shows. The first term corresponds to the scaled Doppler signal, the second term represents the pure scaled noise effect, and the last term accounts for the scaled signal multiplied by both noise and the Doppler effect. Given that \(\epsilon=\frac{a_{2}}{a_{1}}\ll 1\), after filtering and transformation, the effect of the last two terms is reduced by \(\epsilon\) and can be neglected. This shows the importance of introducing a filter before applying the nonlinear transformation.