###### Abstract  
A digital adaptive filtering system is applied to various fields such as current disturbance, noise cancellation, and active vibration and noise control. The least mean squares (LMS) algorithm is widely adopted, owing to its simplicity and low computational burden. A limitation of the LMS algorithm with fixed step size is the trade-off between convergence speed and stability. Several studies have tried to overcome this limitation by varying the step size according to filter input and error; however, the related algorithms with variable step size have not been suitable for signals with complex frequency spectra. As the error decreases, the quality of the output signal deteriorates due to the increase in the higher-order components, depending on the characteristics of the algorithm. Therefore, a novel adaptive filtering algorithm was proposed to overcome these drawbacks. It increased the stability of the system by decreasing the step size using an exponential function. In addition, the error was reduced through normalization using the power of the input signal in the initial state, and the misadjustments in the system were adjusted properly by introducing an energy autocorrelation function of instantaneous error. Furthermore, a novel multi-staged adaptive LMS (MSA-LMS) algorithm was introduced and applied to active periodic structures. The proposed algorithm was validated by simulation and observed to be superior to the conventional LMS algorithms. The results of this study can be applied to active control systems for the reduction of vibration and noise signals with complex spectra in next-generation powertrains, such as hybrid and electric vehicles. Keywords:least mean squares; multi-staged adaptive LMS; active vibration control +
Footnote †: journal: Applied Soft  
autocorrelation function [2]. The variable step-size LMS algorithm (VS-LMS) uses a modified sigmoid function to introduce a new step size with tracking error [3]. A variable-parameter algorithm has been developed in which the step size is determined using the sum of two exponential functions, their coefficients, and the weight ratio of the output error [4]. A modified LMS (MLMS) algorithm with a modified step size, in which the product of filter input and error is used as an exponential decay parameter, has also been developed [5]. Step size is also adjusted by using the linear sum of the step size and error value square of the immediately preceding step. This includes the constraints of the step size range and reduces the sensitivity to misadjustment in a stable or unstable environment [6]. A stochastic gradient adaptive filter with step size varied, using an autocorrelation function of the error has also been developed [7]. A robust adaptive algorithm was developed by using autocorrelation functions of errors, regardless of the presence of disturbances (e.g., noise) [8]. Another adaptive algorithm, developed to effectively cancel the echo of the hands-free telephone, helps estimate the power of an unmeasurable signal using various detection techniques and to derive the optimum step size to make the system faster and more reliable [9]. There are several VS-LMS algorithms, but with increased computation time [10]. Finally, a variable step-size algorithm including filter error, filter input, and an affine projection algorithm has been proposed to realize faster convergence and smaller misadjustments [11]. However, there is little research on achieving stability and rapid convergence in the case of signals with complex frequency spectra, and therefore, a careful analysis of the signal tracking performance depending on the type of signal is needed. Next-generation powertrains, such as electric cars and hybrid cars, exhibit vibration and noise signals with complex spectra associated with motors and gearboxes. However, attenuation of passive vibration through structural changes is significantly limited, and therefore, active control systems and algorithms with new concepts are needed for reduction of vibration and noise.