# A Variable Step Size Normalized Least-Mean-Square Algorithm Based on Data Reuse
## 4 Simulation Results  
Simulations were performed in the context of AEC, in a single-talk scenario [4; 5; 6]. In this case, the microphone (reference/desired) signal results in:  
\[\mathbf{d}(n) =\mathbf{h}^{T}\mathbf{x}(n)+v(n)\] \[=y(n)+v(n), \tag{17}\]  
where \(\mathbf{h}\) contains the coefficients of the acoustic impulse response (of length \(L\)), \(\mathbf{x}(n)\) contains the last \(L\) time samples of the far-end signal (i.e., the input signal), \(v(n)\) is the background noise, and \(y(n)=\mathbf{h}^{T}\mathbf{x}(n)\) represents the echo signal. In this framework, the main goal is to identify the acoustic echo path, \(\mathbf{h}\), with an adaptive filter, \(\widehat{\mathbf{h}}(n)\). In our experiments, it was considered that \(v(n)\) is a white Gaussian noise, so that the echo-to-noise ratio (ENR) was 20 dB. The ENR is defined as \(\sigma_{y}^{2}/\sigma_{v}^{2}\), where \(\sigma_{y}^{2}\) denotes the variance of the echo signal. The acoustic echo path \(\mathbf{h}\) was obtained from a measured acoustic impulse response using \(L=512\) coefficients (depicted in Figure 2a and available at www.comm.pub.ro/plant, accessed on 24 February 2022), and the same length was used for the adaptive filter, using a sampling rate of 8 kHz. Three types of input signals were considered: (i) a white Gaussian noise, (ii) a first-order autoregressive process, referred to as AR(I), which was obtained by filtering a white Gaussian noise through a first-order autoregressive model with a pole at 0.8, and (iii) a recorded speech sequence (depicted in Figure 2b); for a longer simulation time (as needed in some experiments), this voice signal can be simply repeated several times. The experimental framework is summarized in Table 15 in Appendix A. In the middle of some simulations, an echo path change scenario was simulated by shifting the acoustic impulse response to the right by 12 samples, in order to assess the tracking capabilities of the algorithms. The performance measure used in all the experiments was the normalized misalignment (in dB), which is evaluated as:  
\[\text{NM}(n)=20\text{log}_{10}\frac{\left\|\mathbf{h}-\widehat{\mathbf{h}}(n) \right\|}{\left\|\mathbf{h}\right\|}, \tag{18}\]  
where \(\left\|\cdot\right\|\) denotes the Euclidean norm. The parameters of the proposed VSS-NLMS algorithm were set as follows. Based on (16), we set \(k_{\text{max}}=\left\lceil 5/\alpha\right\rceil\), using different values for the lower bound of the normalized step size parameter \(\alpha\). The power estimates from (14) and (15) were computed using \(K=2\) for the white Gaussian input and \(K=6\) for the AR(1) process or speech sequence [14]; also, we used \(Q=3K\) and \(\xi=1.5\)[29]. The main parameters of the algorithms used in comparisons are provided in Table 15 in Appendix A, while their main values are summarized in Table 16 from the same section. First, we evaluated the influence of the parameter \(l\) on the performance of the proposed algorithm (using the lower bound \(\alpha=0.05\)), for different types of input signals. The results are given in Figures 3-5, using white Gaussian noise, the AR(1) process, and the speech signal, respectively. These results supported the discussion from Section 3, concerning the influence of the parameter \(l\) (and the recommended choices). Consequently, in the following simulations, we use \(l=0.1\) for white Gaussian inputs, \(l=0.3\) for the AR(1) process, and \(l=0.5\) for the speech signal. Next, the performance of the VSS-NLMS algorithm was evaluated for different values of the lower bound of the normalized step size parameter, i.e., \(\alpha=0.1,0.05,0.03\), and \(0.01\). The results are given in Figure 6, using a white Gaussian noise as the input. It can be noticed that the algorithm was efficient for different values of \(\alpha\). However, as was explained in Section 3, it is not recommended to choose a very small value of this lower bound.