Novel Adaptive Filtering Algorithms Based on Higher-Order Statistics and Geometric Algebra

Yinmei He,Yi Yan,Xiangyang Wang,Jian Zhou,Rui Wang

doi:10.1109/access.2020.2988521

Abstract

Adaptive filtering algorithms based on higher-order statistics are proposed for multi-dimensional signal processing in geometric algebra (GA) space. In this paper, the proposed adaptive filtering algorithms utilize the advantage of GA theory in multi-dimensional signal processing to represent a multi-dimensional signal as a GA multivector. In addition, the original least-mean fourth (LMF) and least-mean mixed-norm (LMMN) adaptive filtering algorithms are extended to GA space for multi-dimensional signal processing. Both the proposed GA-based least-mean fourth (GA-LMF) and GA-based least-mean mixed-norm (GA-LMMN) algorithms need to minimize cost functions based on higher-order statistics of the error signal in GA space. The simulation results show that the proposed GA-LMF algorithm performs better in terms of convergence rate and steady-state error under a much smaller step size. The proposed GA-LMMN algorithm makes up for the instability of GA-LMF as the step size increases, and its performance is more stable in mean absolute error and convergence rate.

Highlights

Adaptive filtering algorithms (AFs) are based on matrix algebra and vector calculus [1], which do not require prior knowledge of signal statistics
Adaptive filtering algorithms based on higher-order statistics and geometric algebra (GA) are proposed by extending the original least-mean fourth (LMF) and least-mean mixed-norm (LMMN) into GA space, which provide a vectorial representation for multi-dimensional signal and the inherent structures of different components can be totally preserved
In this paper, we propose novel adaptive filtering algorithms, GA-based least-mean fourth (GA-LMF) and GA-based least-mean mixednorm (GA-LMMN), which are based on higher-order statistics and formulates a multi-dimensional signal as a multivector in the GA form

Summary

INTRODUCTION

Adaptive filtering algorithms (AFs) are based on matrix algebra and vector calculus [1], which do not require prior knowledge of signal statistics. Wang et al [30] propose a novel GA-based Least-Mean Kurtosis algorithm (GA-LMK), which improves the convergence behavior of adaptive filter by minimizing the higher-order statisticsbased cost function. In order to further improve the performance of existing GA-based adaptive filtering algorithms without increasing computational complexity, this paper propose GA-LMF and GA-LMMN algorithms. Adaptive filtering algorithms based on higher-order statistics and GA are proposed by extending the original LMF and LMMN into GA space, which provide a vectorial representation for multi-dimensional signal and the inherent structures of different components can be totally preserved. The proposed GA-LMF and GA-LMMN process multi-dimensional signal by minimizing the cost function based on higher-order statistics, which can improve convergence rate and reduce the steady-state error, and the computational complexity is rather lower.

SIMULATION ANALYSIS

COMPUTATIONAL COMPLEXITY

CONCLUSION