## 3 Simulation Results
### Simulation Environment  
All detection techniques were evaluated using MATLAB(r) 2024b simulations to assess their probability of detection (\(P_{d}\)) under varying signal-to-noise ratio (SNR) conditions and phase noise impairments. The input to each simulation was a time-domain beat signal corresponding to a single target, with Gaussian white noise added to simulate different SNR levels ranging from \(-10\) dB to \(+20\) dB in 2 dB steps. Each SNR value was averaged over 1000 Monte Carlo trials to ensure statistical robustness.  
To investigate the impact of oscillator phase noise on detection performance, phase noise was modeled as a multiplicative complex exponential process applied to the beat signal. Four phase noise levels were considered: \(-100\), \(-90\), \(-80\), and \(-70\) dBc/Hz, each corresponding to a realistic low-cost oscillator specification. The noise was added using MATLAB's awgn() function for additive noise and a custom model for phase noise.  
FFT-based processing was performed with a zero-padded size of \(N_{\text{FFT}}=8N\), where \(N\) is the length of the beat signal. A Kaiser window was applied to mitigate spectral leakage. The same FFT configuration was consistently used across all methods, including the subspace-based LTSP and SVD algorithms.  
All simulations were executed using double-precision arithmetic to avoid numerical artifacts, and detection was declared successful if the identified peak index was within a tolerance of \(\pm 3\) bins of the ground truth. The simulation environment was designed to replicate realistic harmonic FMCW radar receiver conditions, providing a fair comparison between lightweight and computationally intensive detection methods.  
Figure 2 illustrates the detailed system model used in the simulations of the harmonic FMCW radar. The system began with a phase-locked loop (PLL) that generated the linear frequency-modulated signal \(s(t)\). This signal was amplified by a transmit amplifier \(G_{T}\) and emitted via a transmit antenna. The transmitted chirp propagated through the environment and interacted with both the intended nonlinear harmonic tag and unwanted reflectors, such as metallic objects or walls, producing multipath components.  
The harmonic tag responded by reflecting a second harmonic signal, which traveled through a multipath channel modeled by multiple delayed reflections. A key component in the model was the short-range leakage, modeled by a fixed delay \(\tau_{S}\) and amplitude gain \(A_{S}\), representing self-interference from the transmit to receive antenna. The received signal was the superposition of harmonic reflections from targets, leakage, and additive white Gaussian noise \(w(t)\).  
This analog signal was then amplified by a low-noise amplifier (LNA) \(G_{L}\) and mixed with a doubled reference signal to produce the intermediate frequency (IF) signal. The mixer output was low-pass filtered using an analog filter \(h_{L}(t)\) to extract the base band component. The resulting signal \(y(t)\) was then sampled and processed using digital peak detection algorithms such as FFT, CA-CFAR, MPM, SVD, and LTSP in the subsequent stages of the simulation.