## 3 Simulation Results of the Proposed Algorithm  
In our simulations, we use the parameters shown in Table 1. The inner workings of the different modules depicted in Figure 1 are detailed in the flowchart presented in Figure 3. On the receiver side, the incoming signal is down-converted by the RF front-end and then digitized by the ADC. A median filter is then applied to the digitized signal. Figure 3(a) depicts a single-pulse Barker-13 with a coherent pulse interval (CPI) at an SNR of \(-5\) dB. After applying the median filter, we obtain the signal shown in Figure 3(b). The figure illustrates the effectiveness of the median filter and its ability to significantly decrease the noise while preserving the edges, including the sharp ones at the points where the signal undergoes a phase flip of \(\pi\). Performing this step is critical for both achieving accurate estimates of the Doppler frequency and minimizing the number of required iterations. \begin{table}
\begin{tabular}{c c c c} \hline
**General Radar Parameters** & **Symbol** & **Value** & **Units** \\ \hline Pulse Width & \(\delta\) & 1.3 & ms \\ Pulse Repetition Interval & \(T_{r}\) & 3.5 & ms \\ Coherent Pulse Interval & CPI & 10 & Scalar \\ Swerlling case & & Swerlling II & \\ Band & & L, S, and C Bands & \\ Selected Carrier Frequency & \(f_{c}\) & 4 & GHz \\ \hline
**Bi-Phase Radar Parameters** & **Symbol** & **Value** & **Units** \\ \hline Sampling frequency & \(f_{s}\) & 120 & MHz \\ Bi-phase type & & Barker-13 & \\
**Target Parameters** & **Symbol** & **Value** & **Units** \\ \hline Target range & \(\tau\) & 30 & km \\ Target Doppler & \(f_{d}\) & 0–15 & kHz \\ CFAR type & & Order-Statistics CFAR & \\ CFAR window size & \(W\) & 8 & Range cells \\ Probability of false alarms & \(P_{fa}\) & \(10^{-6}\) & - \\ \hline
**Noise Parameters** & **Symbol** & **Value** & **Units** \\ \hline Noise type & & Additive White Gaussian Noise (AWGN) & \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 3: A flowchart of the proposed module. In the first iteration, a nonlinear transformation is applied to the filtered waveform. In our algorithm, we chose a quadratic function to remove bi-phase modulation from the signal and to increase the Doppler cycles. The output from this stage is shown in Figure 5. It is worth pointing out that although the median filter was effective in reducing noise and preserving edges, it was not able to maintain sharp transitions in the bi-phase waveform. This was due to the nature of the median filter, which approximated sharp transitions differently. This, in turn, caused distortions in the Euler form at the output of the nonlinear transformation. The effect of these distortions was minor, as the Doppler estimator relied on ensemble averaging in most cases. A moving average filter was employed before the local minmax detector to increase its robustness by smoothing the noisy, close-by peaks, as shown in Figure 6. While it may appear that the MA filter does not make a substantial difference, it is important to note that before applying the moving average in Figure 5, the signal exhibits noticeable oscillations. If these are mistakenly considered as peaks by the proposed system, it would significantly affect the Doppler estimation. Therefore, the function of the moving average is to smooth these oscillations and ensure a more accurate Doppler estimation. Figure 4: Representation of a single pulse of the received signal at SNR = \(-5\) dB (**a**) before filtering. (**b**) Output of the median filter for a single pulse. Figure 5: The output of the nonlinear transformation \(x^{2}\). To improve the clarity, the first \(0.1\) ms segment of the signal has been enlarged. The local minmax detector is analogous to the traditional peak-finding algorithm but is also able to detect valleys. The indices from the local minmax detector are then used for Doppler estimation and phase correction, as discussed in Sections 2.1 and 2.2. Figure 7 shows the signal \(B(t)\) obtained from the Hadamard block in the first iteration.