# An Optimized Differential Step-Size LMS Algorithm
## 4 ODSS-LMS Algorithm
### Practical Considerations
The proposed algorithms are compared with the IPNLMS algorithm [4] (since, in general, it outperforms the NLMS algorithm) and also with the JO-NLMS algorithm [24] (using constant values of the parameter \(\sigma_{w}^{2}\)). In the first experiment, the acoustic echo path is involved. The system is considered time-invariant with \(\sigma_{w}^{2}=0\). The ODSS-LMS-W and ODSS-LMS-G algorithms are compared with the IPNLMS algorithm in Figure 2, where the input signal is a white Gaussian noise. Thus, we confirm that relations (19), (24), and (26) are equivalent to (27), (28), and (29), respectively, in case of white Gaussian noise as an input signal. For the same convergence rate as the IPNLMS algorithm, the normalized misalignment of the proposed algorithms is definitely much lower. Figure 1: Echo paths used in simulations. (**a**) acoustic echo path and (**b**) network echo path (which is adapted from G168 Recommendation [25].) In the context of acoustic echo cancellation, the tracking capability of the algorithm is an important aspect. Unfortunately, the acoustic path can rapidly change during the communication. Therefore, in Figure 3, the ODSS-LMS-G algorithm is analyzed for different values of \(\sigma_{w}^{2}\) in a scenario where the input signal is a white Gaussian noise. The echo path change has been simulated by shifting the impulse response to the right (by 10 samples) in the middle of the experiment. In this scenario, the importance of the variance \(\sigma_{w}^{2}\) is highlighted. The misalignment is much lower for low values of \(\sigma_{w}^{2}\) and is dimmed for high values of this parameter (e.g., \(\sigma_{w}^{2}=10^{-10}\)). Moreover, the variance of system noise indicates the need of a compromise between low misalignment and tracking capability (e.g., for \(\sigma_{w}^{2}=0\), the ODSS-LMS-G algorithm tracks the optimal solution in a slow manner). Figure 3: Normalized misalignment of the IPNLMS and ODSS-LMS-G algorithms for different values of \(\sigma_{w}^{2}\). Echo path changes at time 10 s. The input signal is a white Gaussian noise, \(L=512\), and SNR = 20 dB. The unknown system is the acoustic echo path. Figure 2: Normalized misalignment of the IPNLMS, ODSS-LMS-G, and ODSS-LMS-W algorithms, where \(\sigma_{w}^{2}=0\), the input signal is a white Gaussian noise, \(L=512\), and SNR = 20 dB. The unknown system is the acoustic echo path. Furthermore, we analyze the behavior of the ODSS-LMS-G algorithm for the identification of the sparse network echo path. Figure 4 outlines that the convergence rate is much faster than in the acoustic echo cancellation scenario, and the normalized misalignment decreases with at least 12 dB for all values of \(\sigma_{w}^{2}\) as well. Beyond this, the simulations are performed considering a stationary correlated input signal. In the context of acoustic echo cancellation configuration, Figure 5 shows the performance of the IPNLMS, JO-NLMS, ODSS-LMS-G, and ODSS-LMS-W algorithms in a scenario where the input signal is an AR(1) process, with \(\beta=0.9\). It can be observed that, for high values of \(\sigma_{w}^{2}\), it takes control in Equations (19), (24), (26) and (27)-(29) through \(\boldsymbol{\Gamma}(n)\), and both ODSS-LMS algorithms behave similarly. Next, the ODSS-LMS-G and ODSS-LMS-W algorithms are analyzed for different values of \(\sigma_{w}^{2}\) in a tracking scenario, where the input signal is an AR(1) process, with \(\beta=0.8\). In Figure 6,  
Figure 4: Normalized misalignment of the IPNLMS and ODSS-LMS-G algorithms for different values of \(\sigma_{w}^{2}\). Echo path changes at time 10 s. The input signal is a white Gaussian noise, \(L=512\), and SNR = 20 dB. The unknown system is the network echo path. Figure 5: Normalized misalignment of the IPNLMS, JO-NLMS, ODSS-LMS-G, and ODSS-LMS-W algorithms for different values of \(\sigma_{w}^{2}\). The input signal is an AR(1) process, with \(\beta=0.9\), \(L=512\), and SNR = 20 dB. The unknown system is the acoustic echo path. the unknown system is represented by the sparse network path.