# Design and Implementation of DSLMS Algorithm Based Photoelectric Detection of Weak Signals
## 1 Introduction  
In the actual signal acquisition environment, the effective signals of some scenes could be stronger and accompanied by a large amount of environmental noise. This often makes the acquired signals non-smooth and increases the difficulty of accurately extracting the effective signals from the noisy environment. Researchers have proposed the adaptive filtering technique to improve the signal-to-noise ratio of weak signals in complex noise environments. Reference [1] introduces an adaptive time delay estimation algorithm for low-signal-to-noise-ratio environments that can accurately estimate the filtering results. Reference [2] proposed a method combining LMS and RLS filters, which is able to obtain results close to Kalman filtering performance with low computational complexity. Reference [3] described the principle of active noise reduction and submitted a patent application. Reference [4] proposed a new normalized LMAT (NLMAT) algorithm that outperforms existing algorithms in various noise environments. Reference [5] proposed a fast and stable normalized least mean quadrature (FSNLMF) algorithm with faster convergence. This technique enhances the optimization of the signal-to-noise ratio in the signal-processing process by dynamically adjusting the filtering parameters to adapt to environmental changes. It improves the accuracy and reliability of signal detection in complex noise backgrounds. Therefore, adaptive filtering techniques have been widely used and thoroughly studied in the fields of radar, sonar, communication and navigation systems, etc. [6]. [PERSON] and his colleagues introduced the least mean square error (LMS) algorithm in the early 1980s, simplifying the computational process, reducing the difficulty of implementation, and enhancing adaptability [7]. This algorithm has been applied in numerous fields dueto its superior performance, such as radar beam formation, adaptive interference noise cancelation, and next-generation mobile communication technologies [8; 9; 10; 11]. Traditional least mean square error (LMS) algorithms use a fixed-step-size approach (hereafter referred to as fixed-step-size LMS algorithms), which limits their ability to achieve fast convergence and maintain low steady-state errors. Researchers have conducted extensive studies to address this issue and optimize the performance of fixed-step-size LMS algorithms. A variable-step-size LMS algorithm that employs an S-function to regulate the step size is introduced in the literature [12]. This method dynamically adjusts the step size through the S-function by assigning a more significant step size to quickly approach the optimal solution and decreasing the step size to improve the accuracy as the convergence point approaches. This variable-step-size strategy significantly improves the convergence speed and reduces the steady-state error compared to the fixed-step-size LMS algorithm. However, the steep variation in the S-function near the zero point causes the algorithm to change the step size too fast as it approaches convergence, which increases the error at the steady state of the algorithm. In the literature [13], by applying translation and flip-flop transformations to the S-function, coupled with introducing new parameters to optimize the change at the bottom of the function, the researchers succeeded in improving the performance of the variable-step-size LMS algorithm. Although this approach brought about a performance improvement, the complexity of the algorithm model adversely affected its flexibility. Subsequently, reference [14] explored a new variable-step-size LMS algorithm that draws on the Q-function with the properties of the S-function curve and adjusts the step size using a compensating term inter-correlation function for the relative error. Due to the inherent characteristics of the Q-function, this algorithm faces the same challenge of significant steady-state errors as it approaches the convergence point.