# A Variable Step Size Normalized Least-Mean-Square Algorithm Based on Data Reuse
## 1 Introduction  
The normalized least-mean-square (NLMS) algorithm is the workhorse in many practical applications of adaptive filtering [1, 2, 3]. The main reasons behind its popularity consist of its simple and efficient implementation, together with fairly reasonable performance, in terms of the convergence rate, tracking, and accuracy of the solution. Nevertheless, there is always a need to improve the performance features of the NLMS algorithm, especially in challenging scenarios, e.g., when using highly correlated input signals and/or dealing with long-length adaptive filters. In this framework, a very important application is acoustic echo cancellation (AEC) [4, 5, 6], where the input signal is mainly speech (i.e., a nonstationary and highly correlated sequence), while the impulse response of the acoustic echo path is usually of the order of hundreds or even thousands of coefficients. There are several strategies to improve the performance of the NLMS algorithm in AEC scenarios. Among them, proportionate NLMS algorithms exploit the sparseness nature of the echo paths to increase the convergence rate and tracking of the adaptive filter [7, 8, 9, 10, 11, 12, 13]. Besides, variable step size NLMS (VSS-NLMS) algorithms are designed to achieve a proper compromise between the convergence rate and misadjustment (i.e., accuracy of the estimate) [14, 15, 16, 17, 18]. Moreover, the data reuse approach consists of performingmore than one filter update for the same set of data (i.e., the input and reference signals), thus targeting a higher convergence rate [19; 20; 21; 22; 23; 24; 25]. In this short communication, we focused on the data reuse NLMS algorithm from a different perspective, showing how it can be exploited in order to develop an improved NLMS version with variable step sizes. As compared to other previous works related to the data reuse approach, the proposed VSS-NLMS algorithm uses a sequence of scheduled step size parameters. These values are a priori computed by exploiting the convergence modes of the data reuse NLMS algorithm. On the contrary, in [24], the variable step size (VSS) from [14] was used in conjunction with the data reuse NLMS algorithm from [21], thus resulting in a new version that was similar to an affine projection algorithm with VSS parameter. Furthermore, in [25], the data reuse approach was applied to an optimized least-mean-square algorithm, which behaves similar to a VSS version. In these previous works, the VSS parameter was evaluated within each iteration of the algorithm, which is fundamentally different as compared to the current proposed approach, where the step size parameters were a priori computed and scheduled within the algorithm. Our main application framework was AEC [4], where the main goal was to estimate the impulse response between the loudspeaker and microphone of a hands-free communication device. Basically, we dealt with a system identification problem, which can be solved by using an adaptive filter. Currently, hands-free communication devices are involved in many popular applications, such as mobile telephony and teleconferencing systems. Due to their specific features, they can be used in a wide range of environments with different acoustic characteristics. In this context, an important issue that has to be addressed when dealing with such devices is the acoustic coupling between the loudspeaker and microphone. In other words, besides the voice of the near-end speaker and the background noise, the microphone of the hands-free equipment captures another signal (coming from its own loudspeaker), known as the acoustic echo. Depending on the environments' characteristics, this phenomenon can be very disturbing for the far-end speaker, which hears a replica of her/his own voice. From this point of view, there is a need to enhance the quality of the microphone signal by canceling the unwanted acoustic echo.