# Design and Implementation of DSLMS Algorithm Based Photoelectric Detection of Weak Signals
## 3 DSLMS Adaptive Filter
### Algorithmic Principles  
This study proposes a variable-step-size adjustment strategy based on the minutiae function by taking the inverse of the independent variable (x) of the minutiae function to obtain a new step-size factor-taking function, as shown in Equation (8). \[\mu(x)=1-\frac{1}{1+x^{2}} \tag{8}\]  
where x denotes the autocorrelation estimate based on the current error versus the previous moment's error, bringing \(e_{j}\) from Figure 3 into the equation. From Figure 4a, it can be seen that the \(\mu(\text{x})\) curve conforms to the step-size adjustment principle of the adaptive filtering algorithm in the absence of noise interference or relatively small noise interference in the early stage of the algorithm (the initial stage of algorithmic convergence). The step-size adjustment function provides the algorithm with a more significant value of the step size (\(\mu\)) to improve the convergence speed of the algorithm and to enable it to quickly transit to the stage of the algorithm's convergence completion. In the convergence completion phase, a smaller step size value (\(\mu\)) is offered to keep the algorithm more stabilized in that phase. Figure 4: Convergence performance analysis graph. (**a**) Algorithm convergence performance with no or relatively low noise interference (**b**) Algorithm convergence performance in case of severe noise interference. Figure 3: DSLMS filter structure schematic diagram. However, in the environment with more serious noise interference, as shown in Figure 4b, the error, \(e(n)\), contains the signal component, \(s(n)\), and the residual noise, \(v_{0}(n)-v_{1}(n)\), after being processed by the counter-cancelation system. Despite the use of the cancelation system, if the residual noise is still considerable, it will cause the error, \(e(n)\), to be more significant all the time, so that \(\mu(x)\) cannot go to a minimal value and the adaptive algorithm is too complex to reach the optimal solution. It can only fluctuate around the optimal solution. At this time, the weight-vector step factor must be improved to eliminate the noise interference. \[\mu(n)= \begin{cases}p,&n\leq N_{0}\\ 2\alpha\Bigg{(}\frac{1}{\frac{1}{\beta E(n)e(n-1)^{2}}+1}\Bigg{)}+\gamma,n>N_{0 }\end{cases}  \tag{9}\]  
The DSLMS algorithm proposed in this paper employs an improved weight-vector step-size factor tuning function, shown in Equation (9), which determines the step size based on the current number of iterations, \(n\), and a predefined threshold value, \(N_{0}\). When the number of iterations n is less than or equal to \(N_{0}\), the step size is set to a constant, \(p\). This provides a more significant step size in the initial phase of the algorithm, which speeds up the initial convergence. When the number of iterations \((n)\) exceeds \(N_{0}\), the step-size adjustment becomes a dynamic value dependent on the correlation of the error signal. Fine control of the step size is achieved by introducing the parameters \(\alpha\), \(\beta\), and \(\gamma\). \(\alpha\) is a scaling factor that controls the overall amplitude, determining the baseline size of the step size change. \(\beta\) acts as a scaling factor, which controls how much the correlation of the error signal affects the step size. By adjusting this, we can control how quickly the algorithm responds to dynamic changes in the error signal, which involves the algorithm's adaptability and sensitivity. \(\gamma\) ensures that the algorithm has a minimum step size value, which means that the step size does not drop to zero even when the error correlation is minimal, guaranteeing that the algorithm can continue updating the weights. In summary, the formula for the variable-step-size adaptive filtering algorithm in the adaptive pair cancelation filter proposed in this paper can be summarized as  
\[e_{j}=d_{j}-y_{j}=s(n)+v_{0}(n)-w^{T}(n)x_{j}(n) \tag{10}\]  
where \(e_{j}\) denotes the error signal, the difference between the desired signal and the filter output.