## 3 Simulation Results
### Results
However, the FFT and MPM methods began to show more pronounced degradation, requiring a higher SNR to reach reliable detection. The SVD method remained the least resilient at these noise levels. At high phase noise (\(-70\) dBc/Hz, Figure 8), only CA-CFAR maintained a graceful degradation curve and reached 100% detection at high SNR. Both LTSP and FFT began to struggle below 15 dB SNR, highlighting their vulnerability to strong phase perturbations. Figure 4: Sampled time-domain and frequency-domain representations of the simulated intermediate frequency (IF) signal under severe phase noise of \(-70\) dBc/Hz and SNR = 0 dB. (**a**) Time-domain beat signal \(y_{\text{norm}}[n]\) prior to range compression, heavily corrupted by noise and lacking visible periodicity. (**b**) Frequency-domain magnitude spectrum \(Y_{\text{norm}}[k]\) obtained via FFT, showing significant spectral spreading and side lobes due to phase noise. Despite the challenging conditions, a dominant peak remained observable, allowing for peak-based range estimation. This figure demonstrates the limits of range compression effectiveness under high-phase-noise and low-SNR scenarios. The SVD-based approach failed entirely across the SNR range, indicating its unsuitability in such conditions without additional stabilization mechanisms. These results confirm that while subspace-based methods such as SVD and LTSP provide strong theoretical performance under ideal conditions, traditional methods like CA-CFAR offer superior resilience in the presence of hardware-imposed imperfections such as phase noise. Figure 5: Probability of detection versus SNR for five peak detection algorithms under phase noise of \(-100\) dBc/Hz. The CA-CFAR method achieved the highest robustness across all SNR levels, followed by the proposed LTSP technique. While FFT and MPM performed reasonably well at moderate SNR, SVD-based detection failed to operate reliably below 15 dB due to its sensitivity to phase noise. These results highlight the performance disparity between conventional and subspace-based methods under low-phase-noise conditions. Figure 6: Probability of detection versus SNR under phase noise of \(-90\) dBc/Hz. Compared to the lower phase noise case, all detection algorithms showed a slight performance degradation, with detection curves shifting rightward. CA-CFAR maintained excellent robustness across all SNR levels, while the proposed LTSP continued to outperform FFT and MPM methods at moderate SNRs. The SVD-based approach remained ineffective until the SNR exceeded 13 dB, indicating its sensitivity to spectral distortion in phase-noisy environments. While the proposed LTSP method demonstrated superior detection probability compared to FFT and MPM in mid-SNR regimes, it did not consistently outperform the SVD-based approach in extremely low SNR conditions. This behavior was expected as classical SVD-based detection directly evaluates the singular value structure of the beat signal's Hankel matrix, which, in theory, is effective for low-rank signal detection. However, under heavy noise, the singular values become less distinguishable, leading to performance degradation or delayed detection onset. In contrast, LTSP employs a rank-1 approximation followed by signal reconstruction and spectral peak analysis, which becomes effective once the dominant signal mode is sufficiently above the noise floor, typically in the 5-15 dB SNR range. Therefore, LTSP was intentionally optimized for this practical SNR regime, offering  
Figure 8: Probability of detection versus SNR under phase noise of \(-\)70 dBc/Hz. At this elevated noise level, only CA-CFAR maintained strong detection performance, achieving full detection beyond 10 dB SNR. All other methods, including LTSP and FFT, exhibited a significant rightward shift in their detection curves, requiring much higher SNR to achieve similar reliability. The SVD-based detector failed entirely, confirming its extreme sensitivity to phase noise.