## 1 Introduction  
Pseudorandom binary code-pulse amplitude modulation (PRBC-PAM) signals play a pivotal role in proximity sensors and short-range radar due to their remarkable detection precision. The measurement and estimation of PRBC-PAM signal parameters are crucial for further identification and countermeasure. Nevertheless, these processes are considerably challenging, especially in scenarios where the operational range is extended and the signal-to-noise ratio (SNR) is detrimentally low, which is attributable to the limited power output of the sensor [1; 2; 3]. As depicted in Figure 1, this scenario underscores the significance of accurately measuring and estimating signal parameters under such countermeasure conditions. A lower SNR measured by a reconnaissance system implies the capability to detect and estimate signal parameters from a greater distance relative to the PRBC-PAM sensor. Therefore, studying the measurement and estimation of PRBC-PAM signal parameters is highly important for modern reconnaissance. Although numerous studies on the parameter estimation methods of continuous-wave PRBC signals have been studied, limitations remain, including the general requirement of a known carrier frequency for estimating pseudorandom codes, the low estimation accuracy of the carrier frequency, and poor estimation performance under low SNRs [4; 5; 6; 7]. In contrast, there is not only limited research on pulse PRBC signal parameter estimation methods but also a lack of valuable information. The stochastic resonance (SR) method [8; 9] was used to estimate the carrier frequency for an aperiodic signal and PRBC signal; however, the method cannot estimate the PAM pseudorandom sequence. Ref. [10] presented a dual-channel cooperative sub-Nyquist sampling system for linear frequency-modulated BPSK signals. However, the estimation accuracy of the code phase deteriorates rapidly when the SNR is less than 10 dB. The third-order correlation function method in references [11; 12] was adopted to estimate PAM pseudorandom sequences, but this method requires a known exact carrier frequency. In addition, none of the studies discussed above obtained effective estimation results at low SNRs. In another realm of the literature, the short-time Fourier transform [13], the smoothed pseudo-Wigner distribution [14], and the fractional Fourier transform [15; 16] were used for signal parameter analysis, but the computational cost was large, and the calculation time was high, which made it difficult to fill the real-time demands of signal detection and parameter estimation. By introducing the constant delay, [PERSON] et al. [17] proposed the parameterized centroid frequency-chirp rate distribution (PCFCRD) technique, which is based on a newly defined correlation function, for noisy multicomponent LFM signal analysis. Antinoise performance and estimation results could be obtained at \(-12\) dB SNR. The duffing oscillator is a kind of nonlinear system which has a wide application prospect in the area of weak signal detection owing to its sensitivity to weak periodic signals under the critical chaotic state and resistance to noise [18; 19; 20; 21], which can be very helpful for solving the urgent problem in ultra-low signal-to-noise ratio cases. Its abundant nonlinear properties play a vital role in parameter estimation of the continuous-wave PRBC signal [22; 23; 24; 25; 26], which is the key support to obtain the carrier frequency and pseudorandom sequence estimation at very low SNRs. However, there are few studies about the PRBC-PAM signal parameter estimation method based on the duffing system. Therefore, further research is urgently needed. In view of the outstanding properties of the Duffing oscillator system, a weak signal detection system is proposed by us to obtain the exact frequency and PAM sequence estimation of the weak PRBC-PAM signal when faced with low-SNR environments. The main contributions are as follows:  
1.