# Research on Variable-Step-Size Adaptive Filter Algorithm with a Momentum Term
## 4 Conclusions  
1. Based on the LMS algorithm, a variable-step-size normalization adaptive filter algorithm known as VSNLMS is developed to further minimize convergence errors in adaptive filtering. This algorithm utilizes the logistic function to establish a function model of step size varying with the error. The filtering results of the VSNLMS algorithm are compared to those of the LMS algorithm for wideband signals with noise. It is observed that the VSNLMS algorithm achieves significantly smaller convergence errors. The effects of the VSNLMS algorithm parameters (\(\alpha\), \(\beta\), and \(\gamma\)) on the filtering results are investigated using the control variable method. On this basis, the optimal parameter values are determined.
2. To accelerate the convergence of the algorithm, a momentum term \(\theta\) is introduced into the VSNLMS algorithm, resulting in the modified VSNLMS (M-VSNLMS) algorithm. This term consists of a forgetting factor and a momentum term of weight increment from the previous moment. This modification enables the algorithm to anticipate changes in the error trend, leading to faster convergence. The effects of the M-VSNLMS algorithm parameter \(\theta\) on the filtering results are investigated based on the above conclusion, and the optimal parameter values are determined. Moreover, the filtering results of the M-VSNLMS algorithm are compared to those of the LMS algorithm for wideband signals containing noise. The results demonstrate that the M-VSNLMS algorithm achieves signal reconstruction and effective noise suppression in the time domain, frequency domain, and energy distribution. Furthermore, it exhibits a significantly faster convergence rate compared to that of the LMS algorithm.
3. The M-VSNLMS algorithm is utilized to perform system identification on a noisy dynamic system mathematical model. The results show that the proposed algorithm can accurately identify the mathematical model with an accuracy of over 97%, while also suppressing the noise components in the logarithmic model by over 99%. Real-time vibration data fitting experiments were conducted in the aeroelastic laboratory of CARDC. The data fitting capabilities and mathematical model complexity of four algorithms, ARMAX, N4 SID, LMS, and M-VSNLMS, were compared, indicating that M-VSNLMS has a higher real-time data fitting accuracy and lower-complexity mathematical models.  
The weight update of the algorithm relies on the error gradient and the magnitude of the step size. However, for strongly nonlinear and intermittently active systems, the algorithm may encounter the local optimum of the error gradient or large fluctuations in the step size, leading to prolonged convergence or even divergence. To address this issue, further analyses are necessary regarding the computation and processing of the system input data in higher-dimensional spaces, so as to enable the identification and noise reduction of more complex systems. Notably, recent research has proposed algorithms such as Kernel Adaptive Filtering (KAF) and Convolutional Neural Network (CNN), which offer valuable insights in this domain.  
Writing--original draft preparation, B.L. ([PERSON]); resources, B.L. ([PERSON]); project administration, X.K.; data curation, Y.S.; formal analysis, L.Y.; funding acquisition, H.G. and B.L. ([PERSON]); methodology, K.Z. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The study did not report any data.  
The authors declare no conflict of interest.