# Design and Implementation of DSLMS Algorithm Based Photoelectric Detection of Weak Signals
## 3 DSLMS Adaptive Filter
### Algorithm Performance Analysis  
To better analyze the specific impact of the parameters in the step adjustment function model on the algorithm's performance, this paper will focus on the volatility of the step coefficients as well as the role of parameters \(\alpha\) and \(\beta\) on the convergence speed of the algorithm. On this basis, we will discuss in detail the selection principle of each parameter and its scope of application. When analyzing the step-size adjustment strategy, we will choose the specific condition of \(\gamma\) = 0 to explore the influence of \(\alpha\) and \(\beta\) parameters. Figure 5a shows the step coefficient, \(\mu(n)\), and error \(\big{(}\,e_{j}\,\big{)}\) adjustment curve when \(\alpha\) is different and \(\beta\) is the same; Figure 5b shows the step coefficient, \(\mu(n)\), and error \((e_{j})\) adjustment curve when \(\alpha\) is the same and \(\beta\) is different; and Figure 6 shows the fluctuation in algorithmic weight vectors when the parameters \(\alpha\) and \(\beta\) are fixed and \(\gamma\) is taken to other values. The three curves in Figure 5a are the corresponding step-size adjustment curves when \(\alpha\) is taken as 0.5, 1, and 2 (\(\beta\) = 0.010), respectively, from which it can be seen that the larger the value of \(\alpha\), the larger the value of step size provided for the algorithm in the early stage of convergence, but the algorithm decreases in stability in the stage of convergence completion. The smaller the value of \(\alpha\), the more stable the algorithm is in the stage of convergence completion, but the algorithm's speed of convergence is slower; because of the equilibrium stability and the speed of convergence, \(\alpha\) = 2 is taken in this paper. The three curves in Figure 5b are the corresponding step-size adjustment curves when \(\beta\) is 0.01, 0.02, and 0.05 (\(\alpha\) = 2), respectively. It can be seen that the smaller the value of \(\beta\), the smaller the change in the step-size factor in the error close to zero, but in the early stage of algorithm convergence, the algorithm is not able to provide a more significant value of step size and it slower; when \(\beta\) is taken with a more substantial value, it can provide the algorithm with a more significant value of step size to make it converge more slowly. When the value of \(\beta\) is more critical, it can give the algorithm a faster convergence speed. Still, at the same  
Figure 5: Effect of \(\alpha\) and \(\beta\) on the convergence speed and stability of the algorithm: (**a**) \(\beta\) is the same and \(\alpha\) is different; (**b**) \(\alpha\) is the same and \(\beta\) is different. Figure 6: Fluctuation of weight vectors of the \(\gamma\)-different-time algorithm. time, the step factor will be affected by the error when the error is close to zero. The value of \(\beta\) should be in the range of 0.005-0.500; the value of \(\beta\) is taken as \(\beta\) = 0.01 in this paper, and the algorithm's performance in terms of convergence speed and steady-state error performance is better. Figure 6 illustrates the effect of the parameter \(\gamma\) on the convergence performance of the algorithm's weight vectors. A higher value of \(\gamma\) accelerates the convergence of the weight vectors in the early stages of the algorithm. Still, it leads to more significant fluctuations in the weight vectors when the algorithm reaches the steady state. On the contrary, a lower value of \(\gamma\) makes the convergence speed slower but is conducive to reducing the fluctuation at the steady state and enhancing the algorithm's stability. Therefore, choosing the \(\gamma\) value requires a trade-off between convergence speed and strength, depending on the requirements of the application scenario. In this study, after comparative analysis and considering the convergence speed and weight fluctuation at steady state, we choose \(\gamma\) = 0.2 to ensure that the algorithm achieves good noise suppression while maintaining reasonable performance.