Geometric algebra and cosine-function based variable step-size adaptive filtering algorithms

Adaptive filtering algorithms are currently successfully employed in a number of fields.But, a disadvantage of traditional real-valued fixed step-size adaptive filtering algorithm is that it is unable to meet both requirement of convergence rate and the steady-state error.In this article, we presented cosine function and geometric algebra(GA) based variable step-size technique for adaptive filtering.Initially, a multi-dimensional signal is represented as a GA multi-vector for the vectorization process in the proposed approach of adaptive filtering with variable step-size based on GA.Then, by establishing a non-linear function relationship between error signal e(n) and the step-size factor \(\mu \) $\mu$ , given method of adaptive filtering resolves the contradiction among steady-state error and the convergence rate.Finally, simulation results illustrate that in comparison with other existing adaptive filtering algorithms, the defined approach performs better in terms of convergence rate, steady-state error, robustness against impulsive noise and the computational complexity.