A novel approach to geometric algebra-based variable step-size LMS adaptive filtering algorithm

The study of signal processing has recently devoted significantly more attention to adaptive filtering techniques. By addressing the shortcoming of the conventional geometric algebra-based fixed step-size least mean square algorithm that is unable to satisfy in terms of both reducing steady-state error and a faster convergence speed, simultaneously, this study presents an improved logarithmic function-based variable step-size least mean square geometric algebra adaptive filtering algorithm by establishing the step-size factor \(\mu \) $\mu$ and error signal e(n) nonlinear function relationship. The instantaneous values of a current error estimate e(n) and the previous error estimate \(e(n-1)\) $e(n-1)$ are used to determine the step size of the defined algorithm. Besides, an extensive discussion is given on the performance of algorithm under influence of parameters \(\varvec{\gamma }\) $\mathit{\gamma }$ and \(\varvec{T}\) $\mathit{T}$ as well as comparative analysis with other existing geometric algebra-based adaptive filters. Computer simulation reveals that the proposed approach not only has a low steady-state error, robustness against impulsive noise, and fast convergence speed, but it also overcomes some existing algorithm’s instability under steady-state phase.