# Detection of Weak Radar Return Signals Based on Pseudo-Time Domain Algorithm
## 2 Introduction to Denoising Methods  
This study is entirely based on signal spectra. It defines the frequency domain as a \"pseudo-time domain\", where the frequency domain features of the detected signals, including but not limited to signal spectra and power spectra, are regarded as waveforms in this pseudo-time domain. Specific structures within the pseudo-time domain waveforms,corresponding to specific frequency domain features, are identified using waveform detection methods. Therefore, the subject of this research is the frequency domain waveform of signals. After converting time-domain signals into frequency domain signals, denoising and detection of received signals can also be carried out in the \"pseudo-time domain\" using time-domain denoising methods. The chosen denoising method in this paper is the Discrete Wavelet Transform (DWT) [27]. The DWT allows for multiscale analysis of signals, enabling the localization, detection, and extraction of useful signals within a specific range. In this paper, the entire process of denoising and detecting weak return signals is illustrated in Figure 1, which involves five steps. **Step 1:** The input pseudo-time domain signals come in three forms. Background noise, which occurs when the transmitter is off and the receiver detects noise, is the sum of spatial environmental noise, system thermal noise from the receiver, and other external interferences. This signal is in the time domain, with its corresponding frequency domain signal denoted as pseudo-time domain signal A. The reference signal is the initially preset signal from the transmitter, also in the time domain, with its frequency domain signal represented as pseudo-time domain signal B. The received signal, which may or may not contain the transmission signal, is the signal received by the receiver when the transmitter  
Figure 1: Flow chart of signal denoising. is turned on. This signal is in the time domain, with its frequency domain signal referred to as pseudo-time domain signal C. Upon inputting pseudo-time domain signals A, B, and C, the range of SNR for the signal to be detected is determined, and filtering is applied. **Step 2:** Based on the differences in pseudo-time domain signals A and B, wavelet bases and wavelet decomposition levels are selected to decompose the pseudo-time domain signals. This decomposition yields wavelet coefficients corresponding to different decomposition scales in the pseudo-time domain. Traditional wavelet decomposition is performed in the time domain, but in Step 2, the decomposition is conducted in the pseudo-time domain. Methods to determine the decomposition level \(n\) include, but are not limited to, the following approaches:  
\[log_{2}\left(\frac{\frac{F_{k}}{2}}{f}\right)<n<log_{2}\left(\frac{\frac{F_{k} }{2}}{f}\right)+1 \tag{1}\]  
\(n\), representing the wavelet decomposition level, is an integer value determined by the receiver's sampling rate and the signal frequency. **Step 3:** Select the \(i\)-th level wavelet coefficients corresponding to the input pseudo-time domain signals A, B, and C. Denoise this level of coefficients, resulting in updated wavelet coefficients. Traditional thresholding methods include three types: hard thresholding [28], soft thresholding [29], and forced denoising [30; 31; 32]. In this paper, select the \(i\)-th level wavelet coefficients \(A_{1}\), \(B_{1}\), and \(C_{1}\) corresponding to pseudo-time domain signals A, B, and C. Based on Equations (2)-(4), the coefficients \(d_{i1,k}\) at this level are denoised to obtain the updated wavelet coefficients \(d_{i2,k}\), Therefore, the updated \(A_{i2}\), \(B_{i2}\), and \(C_{i2}\) are obtained. The denominator \(a\) is determined by analyzing multiple sets of background noise at this layer, which influences the detection signal accuracy. Here, \(\sigma_{n}\) denotes the global noise standard deviation, and \(\sigma_{i}\) represents the noise standard deviation of the background noise at the \(i\)-th layer.