# Research on Variable-Step-Size Adaptive Filter Algorithm with a Momentum Term
## 3 Numerical Simulation and Verification  
### Determine the Optimal Algorithm Parameters  
The standard wideband signal \(x(t)\) is obtained as follows:  
\[x(t)=\exp(i\times 2\pi\times(f_{0}t+\frac{1}{2}kt^{2})) \tag{14}\]  
where \(f_{0}\) is the starting frequency and \(k\) is the rate of frequency change over time. Setting \(f_{0}\) to 10 Hz, \(k\) to 500, and the sampling period to 0.001 s, the standard LMS algorithm and the variable-step normalization LMS (VSNLMS) algorithm are used to filter the noisy signal \(\mathsf{s}(t)\). The filtered signals \(\mathsf{s}^{\prime}(t)\) and \(x(t)\) are compared for their errors. Figure 3a shows the time domain of the original signal \(x(t)\) and the noisy signal \(\mathsf{s}(t)\) with white noise, and Figure 3b shows the Hilbert time-frequency energy. As shown in Figure 4, the output error diagram of a filter pair based on the LMS algorithm and VSNLMS algorithm is shown. Obviously, the output error of the VSNLMS algorithm is significantly smaller than that of the LMS algorithm, indicating that the VSNLMS outperforms the LMS in filtering. The Pearson correlation coefficient \(R\) between \(\mathsf{s}^{\prime}(t)\) and \(x(t)\) is calculated to evaluate the filtering performance of different algorithms. Moreover, to quantify the impact of parameter changes on the filtering performance of the VSNLMS algorithm, the control variable method is used to obtain the \(R\) value between \(\mathsf{s}^{\prime}(t)\) and \(x(t)\) when the parameters \(\alpha\), \(\beta\), and \(\gamma\) range from 0.1 to 0.9. Finally, the optimal filtering parameters for the VSNLMS algorithm are determined. The formula for calculating the Pearson correlation coefficient is obtained as follows:  
\[R=\frac{\sum\limits_{i=1}^{n}{(x_{i}-\overline{x})(y_{i}-\overline{y})}}{\sqrt {\sum\limits_{i=1}^{n}{(x_{i}-\overline{x})^{2}\sqrt{\sum\limits_{i=1}^{n}{(y_ {i}-\overline{y})^{2}}}}}} \tag{15}\]  
where \(x_{i}\), \(y_{i}\), \(\overline{x}\), and \(\overline{y}\) represent the sample points and the corresponding means for the two datasets. Table 1 shows the R values obtained based on the LMS and VSNLMS algorithms. Figure 4: Schematic diagram of filtering error based on LMS and VSNLMS algorithms. Figure 3: Time domain and Hilbert time-frequency energy of original signal and noisy signal. According to the above quantitative analysis, the optimal parameters for the VSNLMS algorithm are as follows: \(\alpha=0.9\), \(\beta=0.9\), and \(\gamma=0.1\). On this basis, the impact of \(\theta\) on the M-VSNLMS algorithm is further analyzed. Table 2 reports the R values obtained by the LMS and M-VSNLMS algorithms. By adjusting the value of \(\theta\), the variation trend of \(R\) with \(\theta\) is investigated based on the conclusions in Table 1.  
It can be observed from Tables 1 and 2 that as the parameters \(\alpha\), \(\beta\), and \(\theta\) increase, the correlation coefficient \(R\) between the filter output sequence and the original sequence also increases, and the algorithm diverges when \(\theta\) is set to 0.8 or 0.9. Nevertheless, the correlation coefficient \(R\) declines as the parameter \(\gamma\) increases. Figure 5 shows the relationship between the Pearson correlation coefficient \(R\) and values of the parameters \(\alpha\), \(\beta\), \(\gamma\), and \(\theta\) obtained via the control variable method. From Figure 5, it can be observed that the correlation coefficient between the output sequence of the M-VSNLMS algorithm and the original signal is much higher than the result of the LMS algorithm. Additionally, parameters \(\alpha\) and \(\beta\) have the most significant impact on the algorithm, and the momentum term coefficient \(\theta\) further improves the prediction