## 3 Simulation Results
### Results  
Figure 3 presents the time- and frequency-domain representations of a simulated intermediate frequency (IF) signal under adverse operating conditions, specifically phase noise of \(-100\,\mathrm{dBc}/\mathrm{Hz}\) and an SNR of \(0\,\mathrm{dB}\). As shown in Figure 3a, the time-domain signal \(y_{\mathrm{norm}}[n]\) exhibited significant amplitude fluctuations and noise contamination, which obscured the underlying beat waveform. This challenges conventional time-domain detection strategies. However, Figure 3b demonstrates the effectiveness of FFT-based range compression, where the frequency-domain spectrum \(Y_{\mathrm{norm}}[k]\) reveals a dominant peak corresponding to the target beat frequency, despite the low SNR and spectral distortion. This confirms that even in the presence of severe phase noise, the spectral peak remained detectable after range compression, thereby validating the robustness of the detection pipeline employed in this work. Figure 2: System model used in the simulations for harmonic FMCW radar. The radar transmitted a chirp signal \(s(t)\) through a power amplifier and received the second harmonic signal reflected by a nonlinear tag. The backward channel included multipath propagation (modeled by delays \(\tau_{1},\tau_{2},\tau_{3}\)) and additive white Gaussian noise \(w(t)\). After low-noise amplification, the signal was mixed with a doubled reference signal, filtered, digitized, and passed to the peak detection stage. Figure 4 presents the time-domain and frequency-domain representations of the simulated intermediate frequency (IF) signal under severe phase noise of \(-70\,\mathrm{d}\mathrm{B}\mathrm{c}\mathrm{/}\mathrm{H}\mathrm{z}\) and an SNR of \(0\,\mathrm{d}\mathrm{B}\). As shown in Figure 4a, the time-domain beat signal \(y_{\mathrm{norm}}[n]\) was highly corrupted by noise, exhibiting significant amplitude fluctuation and no clear sinusoidal structure. This illustrates the difficulty of extracting range information directly from the time-domain signal under such degraded conditions. In contrast, Figure 4b demonstrates the benefit of FFT-based range compression. The resulting frequency-domain magnitude spectrum \(Y_{\mathrm{norm}}[k]\) showed a broadened main lobe and noticeable spectral spreading due to the phase noise, yet a dominant peak remained discernible. This confirms that despite the adverse conditions, the spectral peak used for range estimation could still be reliably identified, highlighting the resilience and limitations of FFT-based processing in extreme noise scenarios. Figure 3: Sampled time-domain and frequency-domain representations of the simulated intermediate frequency (IF) signal under challenging conditions of phase noise \(-100\,\mathrm{d}\mathrm{B}\mathrm{c}\mathrm{/}\mathrm{H}\mathrm{z}\) and \(\mathrm{SNR}=0\,\mathrm{d}\mathrm{B}\). (**a**) Time-domain beat signal \(y_{\mathrm{norm}}[n]\) prior to range compression. (**b**) Frequency-domain magnitude spectrum \(Y_{\mathrm{norm}}[k]\) after FFT-based range compression, revealing a clearly observable spectral peak despite the high noise level. Figures 5-8 illustrate the probability of detection versus SNR for the five evaluated peak detection algorithms under increasing levels of phase noise: \(-100\), \(-90\), \(-80\), and \(-70\) dBc/Hz, respectively. As expected, all techniques demonstrated degraded performance as the phase noise became more severe. At low phase noise (\(-100\) dBc/Hz, Figure 5), CA-CFAR and LTSP offered superior robustness, achieving near-perfect detection for SNRs above 0 dB. The MPM and FFT methods exhibited acceptable performance, with detection probability exceeding 90% at SNRs above 10 dB, whereas the SVD-based method showed a delayed onset in detection due to its sensitivity to phase distortion. At moderate phase noise levels (\(-90\) and \(-80\) dBc/Hz in Figures 6 and 7), LTSP continued to outperform other techniques, maintaining detection above 90% at moderate SNRs. CA-CFAR demonstrated high reliability and shifted only slightly with increasing noise.