# Research on Variable-Step-Size Adaptive Filter Algorithm with a Momentum Term
## 3 Numerical Simulation and Verification
### Determine the Optimal Algorithm Parameters
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**Based on LMS Algorithm**} \\ \hline
**R1** & & \multicolumn{8}{c}{0.8822} \\ \hline  & & \multicolumn{8}{c}{**VSNLMS algorithm**, \(\beta\), \(\gamma\)** **take 0.5**, \(\alpha\)** **take 0.1**–**0.9**} & \\
**R2** & 0.9505 & 0.9656 & 0.9718 & 0.9752 & 0.9774 & 0.9789 & 0.9801 & 0.9810 & 0.9817 \\ \hline  & & \multicolumn{8}{c}{**VSNLMS algorithm**, \(\alpha\), \(\gamma\)**take 0.5**, \(\beta\)** **take 0.1**–**0.9**} & \\
**R3** & 0.9507 & 0.9655 & 0.9716 & 0.9750 & 0.9772 & 0.9787 & 0.9799 & 0.9808 & 0.9815 \\ \hline  & & \multicolumn{8}{c}{**VSNLMS algorithm**, \(\alpha\), \(\beta\)**take 0.5**, \(\gamma\)** **take 0.1**–**0.9**} & \\
**R4** & 0.9779 & 0.9778 & 0.9777 & 0.9776 & 0.9775 & 0.9774 & 0.9773 & 0.9772 & 0.9771 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Changes in correlation coefficient R values of LMS and VSNLMS algorithms under different parameter conditions. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**LMS Algorithm**} \\ \hline
**R1** & \multicolumn{8}{c}{0.8822} \\ \hline  & & \multicolumn{8}{c}{**M-VSNLMS algorithm**, \(\alpha\), \(\beta\) take 0.9, \(\gamma\)**take 0.1**, \(\theta\)**take 0.1**–**0.9**} & \\
**R5** & 0.9778 & 0.9792 & 0.9796 & 0.9811 & 0.9821 & 0.9831 & 0.9835 & NaN & NaN \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Changes in correlation coefficient R values of LMS and M-VSNLMS algorithms under different parameter conditions. Figure 5: The correlation between R and parameters \(\alpha\), \(\beta\), \(\gamma\), and \(\theta\). accuracy of the algorithm. Although the increase in \(\gamma\) results in a decrease in prediction accuracy, this impact is slight. The above analysis shows that the optimal parameters for the M-VSNLMS algorithm are as follows: \(\alpha=0.9\), \(\beta=0.9\), \(\gamma=0.1\), and \(\theta=0.7\). On the basis of the above conclusions, the LMS and M-VSNLMS algorithms are used to filter the noisy signal \(\mathrm{s}(t)\). Figure 6 illustrates the convergence process of the weight values for both algorithms. For the LMS algorithm, since its weight increment is fixed-step, the weight convergence curve is smooth, but the convergence rate is obviously slow. And at the sampling time of 15 s, the weights are still in a slow iteration process. For the M-VSNLMS algorithm, since its weight increment is variable-step, the weight convergence curve is erratic, but the convergence rate is fast. At t1 = 1.5 s, the weight of each tap quickly converges, and at t1 = 4 s, the weight of each tap roughly converges to a stable state. These demonstrate that the M-VSNLMS algorithm not only has significantly smaller convergence errors compared to those of the LMS algorithm but also has a distinct advantage in terms of the convergence rate. Figure 7 presents the performance of the LMS, VSNLMS, and M-VSNLMS algorithms in filtering the signal \(\mathrm{s}(t)\). The results demonstrate that the M-VSNLMS algorithm has the best performance on original signal reconstruction and noise removal in the time domain, frequency domain, and energy distribution. Figure 7 shows the Hilbert time-frequency of the \(\mathrm{s}(t)\) filtering results using different algorithms. Figure 6: Weight convergence process of LMS and M-VSNLMS algorithms. Figure 7: Hilbert time-frequency plots of S(t) filtering results using different algorithms.