# Detection of Weak Radar Return Signals Based on Pseudo-Time Domain Algorithm
## 3 Experimental Results and Analysis  
### Simulation Experiment Results and Discussion  
The received signals used in this paper include Gaussian white noise with a single-frequency sine signal and Gaussian white noise without it. The expression for the received signal is as follows:  
\[\begin{split} x_{1}(t)&=n(t)+s(t)\\ x_{2}(t)&=n(t)\end{split} \tag{5}\]  
where \(s(t)\) is the single-frequency sine signal, and \(n(t)\) is the Gaussian white noise.  
Parameters are set as follows: the frequency of the sine signal is \(F=50\) Hz, the SNR is set to SNR \(=-24\) dB. Based on the SNR, zero-mean Gaussian white noise is added to the sine signal. The sampling frequency is \(F_{\rm s}=1000\) Hz, and the number of sampling points is \(N=4096\).  
From Figure 2, it can be observed that at a SNR of \(-24\) dB, the sine signal is completely masked by noise in both the time domain and the pseudo-time domain. In the pseudo-time domain, the noise intensity exceeds that of the signal. The distinction between the received signal with the added sine signal and the background noise signal is almost imperceptible. Therefore, under low SNR conditions, direct signal identification is not possible, and the task of signal detection and extraction becomes highly challenging.  
For signals with a specific frequency, the bandwidth of the band-pass filter is set to \(2d\) Hz, so the frequency range of the received signal after filtering is \((F-d,F+d)\) Hz. In this study, the received signals after preprocessing are in the frequency bands of \((-60,\,-40)\) Hz and \((40,60)\) Hz. The signals used in Figure 2a,c are the received signals containing the sinusoidal signal spectrum, while those in Figure 2b,d are the received signals without the sinusoidal signal spectrum. The SNR of the received signal changes from approximately \(-24\) dB to approximately \(-9\) dB after filtering in Figure 3a,b. The received signal after filtering is subjected to wavelet decomposition and denoising, and the reconstructed waveform after denoising is compared with the original received signal waveform. After denoising, the SNR of the received signal is significantly increased, as shown in Figure 3c,d.  
A single set of data cannot adequately validate the accuracy of the method. Therefore, this paper performs denoising on 1000 sets of received signals multiple times, compares the results with the transmission conditions, and calculates the detection probability to verify the effectiveness and feasibility of the method.  
Figure 2: Waveforms of received signals at SNR \(=-24\) dB. (**a**) Time-domain waveform of signal \(x_{1}(t)\); (**b**) Time-domain waveform of signal \(x_{2}(t)\); (**c**) Pseudo-time-domain waveform of signal \(x_{1}(t)\); (**d**) Pseudo-time-domain waveform of signal \(x_{2}(t)\).  
The relationship between SNR and detection rate is shown in Figure 4. It can be seen that within the range of SNR \(\geq-34\) dB, the detection accuracy is 100%. Within the range of SNR \(<-34\) dB, the detection rate decreases as the SNR decreases. When the SNR is \(-40\) dB, the detection success rate can exceed 95%. Therefore, the simulation results can verify that the method adopted in this paper is an effective denoising method.  
Figure 4: Relationship between SNR and detection rate. In the figure, TP represents the probability of correctly identifying the presence of a signal. TN represents the probability of correctly identifying the absence of a signal. FP represents the false alarm probability, which is the probability of identifying the presence of a signal when there is none. FN represents the miss probability, which is the probability of identifying the absence of a signal when there is one. TP + TN represents the total probability of correctly identifying received signals.  
Figure 3: Signal waveforms of received signals before and after denoising. (**a,b**) show the signal waveforms before wavelet denoising, while (**c,d**) show the signal waveforms after wavelet transformation.