Geometric-algebra affine projection adaptive filter

Yuetao Ren,Yongfeng Zhi,Jun Zhang

doi:10.1186/s13634-021-00790-y

Abstract

Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals.

Highlights

With the development of sensor technology, there are more and more data sources for recording the same process
In geometric algebra (GA)-based algorithms, the hypercomplex signals are transformed into multivectors, such as complex entries, quaternion entries, and higher dimensional entries [3], and handled holistically [4]
5 Conclusions The GA-affine projection algorithm (APA) and the R-GA-APA proposed in this article have improved estimation capabilities with highly colored input signals for hypercomplex processes

Summary

Introduction

With the development of sensor technology, there are more and more data sources for recording the same process. The electromagnetic vector-sensor consists of 6 spatially arranged antennas, which measure the electric and magnetic field signals in the three directions of the incident wave [2]. These signals derive from observations of different dimensions. They are constructed into vectors and processed as multi-channel signals in most existing literature. In geometric algebra (GA)-based algorithms, the hypercomplex signals are transformed into multivectors, such as complex entries, quaternion entries, and higher dimensional entries [3], and handled holistically [4]. Owing to the convenience of GA-based models, GA has been studied in many applications, such as classification, direction of arrival estimation, and image processing [7,8,9]

Methods

Results

Conclusion