# A Variable Step Size Normalized Least-Mean-Square Algorithm Based on Data Reuse
## 4 Simulation Results
Figure 2: Experimental framework for acoustic echo cancellation: (**a**) measured acoustic impulse response used in the simulations, with \(L=512\); (**b**) recorded speech signal used in some of the experiments. Figure 4: Misalignment of the NLMS algorithm with \(\alpha=1\) and \(\alpha=0.05\) and misalignment of the VSS-NLMS algorithm using different values of the parameter \(l\) and the lower bound \(\alpha=0.05\). The input signal is an AR(1) process, \(L=512\), and ENR \(=20\) dB. Figure 3: Misalignment of the NLMS algorithm with \(\alpha=1\) and \(\alpha=0.05\) and misalignment of the VSS-NLMS algorithm using different values of the parameter \(l\) and the lower bound \(\alpha=0.05\). The input signal is white and Gaussian, \(L=512\), and ENR \(=20\) dB. Figure 5: Misalignment of the NLMS algorithm with \(\alpha=1\) and \(\alpha=0.05\) and misalignment of the VSS-NLMS algorithm using different values of the parameter \(l\) and the lower bound \(\alpha=0.05\). The input signal is speech, \(L=512\), and \(\text{ENR}=20\) dB. Figure 6: Misalignment of the NLMS algorithm with different values of \(\alpha\) and misalignment of the VSS-NLMS algorithm using \(l=0.1\) and different values of the lower bound for the normalized step size parameter. The input signal is white and Gaussian, \(L=512\), and \(\text{ENR}=20\) dB. In the following experiments, the nonparametric VSS-NLMS (NPVSS-NLMS) algorithm proposed in [14] is introduced for comparison, assuming that the true power of the system noise, \(\sigma_{v}^{2}\), is available for this algorithm. The update of the NPVSS-NLMS algorithm is:  
\[\widehat{\mathbf{h}}(n)=\widehat{\mathbf{h}}(n-1)+\frac{\widetilde{\mu}(n) \mathbf{x}(n)e(n)}{\mathbf{x}^{T}(n)\mathbf{x}(n)}, \tag{19}\]  
where:  
\[\widetilde{\mu}(n)=\left\{ \begin{array}{ll}1-\frac{\sigma_{v}^{2}}{\widetilde {\sigma}_{e}^{2}(n)},&\text{if }\widehat{\sigma}_{e}^{2}(n)\geq\sigma_{v}^{2}\\ \\ 0,&\text{otherwise}\end{array} \right. \tag{20}\]  
and \(\widehat{\sigma}_{e}^{2}(n)\) was evaluated similar to (14). Furthermore, a very small positive constant should be added to the denominator of \(\widetilde{\mu}(n)\), in order to avoid any potential division by zero related to the estimation of \(\widehat{\sigma}_{e}^{2}(n)\). It should be outlined that the NPVSS-NLMS algorithm is usually considered as a benchmark (for many VSS-NLMS algorithms), due to its good performance and practical features. In the following experiments, the proposed VSS-NLMS algorithm uses the lower bound value \(\alpha=0.03\). In order to test the tracking capabilities of the algorithms, an echo path change scenario was considered. All three types of input signals were used, i.e., white Gaussian noise, the AR(1) process, and speech; the results are provided in Figures 7-9, respectively. It can be noticed that the NPVSS-NLMS algorithm outperformed the proposed VSS-NLMS algorithm in the case of the white Gaussian input. However, the proposed algorithm performed better for the AR(1) process and speech inputs. Figure 7: Misalignment of the NLMS algorithm with \(\alpha=1\) and \(\alpha=0.03\), misalignment of the NPVSS-NLMS algorithm, and misalignment of the VSS-NLMS algorithm using \(l=0.1\) and the lower bound \(\alpha=0.03\). The input signal is white and Gaussian, \(L=512\), and \(\text{ENR}=20\) dB, and the echo path changes at time \(15\) s.  
Figure 8: Misalignment of the NLMS algorithm with \(\alpha=1\) and \(\alpha=0.03\), misalignment of the NPVSS-NLMS algorithm, and misalignment of the VSS-NLMS algorithm using \(l=0.3\) and the lower bound \(\alpha=0.03\). The input signal is an AR(1) process, \(L=512\), and \(\text{ENR}=20\) dB, and the echo path changes at time 60 s.  
Figure 9: Misalignment of the NLMS algorithm with \(\alpha=1\) and \(\alpha=0.03\), misalignment of the NPVSS-NLMS algorithm, and misalignment of the VSS-NLMS algorithm using \(l=0.5\) and the lower bound \(\alpha=0.03\).