Elsevier

Signal Processing

Volume 178, January 2021, 107810
Signal Processing

Short communication
Quaternion kernel recursive least-squares algorithm

https://doi.org/10.1016/j.sigpro.2020.107810Get rights and content

Abstract

Various kernel-based algorithms have been successfully applied to nonlinear problems in adaptive filters over the last two decades. In this paper, we study a kernel recursive least squares (KRLS) algorithm in the quaternion domain. By the generalized Hamilton-real calculus method, we can apply the kernel trick to calculate the quaternion KRLS filter. In order to show the feasibility of the proposed algorithm, firstly we investigate the quaternion recursive least squares (QRLS) algorithm, and simulations show that the proposed QRLS algorithm has the same steady error as that of the closed-form solution; Secondly, we generalize the QRLS algorithm to the quaternion KRLS algorithm, theoretical analysis show the convergence, and simulations are described demonstrating the performance of the proposed algorithm.

Introduction

The quaternion algebra describes a wide variety of geometrical objects (such as points, lines, and planes), simplifies many physical problems [1], [2], and have been widely used in various fields, such as quaternion neural network [3], three-dimension motion tracking [4], four-dimension biomedical signal processing [5], frequency domain adaptive filtering [6] and nonlinear adaptive system using kernel adaptive filters (KAFs) [7]. In the quaternion domain, the mutual information among all four components can be performed with quaternion multiplication. The 4-D signals in the real domain cannot incorporate the mutual information [7]. The mutual information may help reduce the complexity. It is shown in [3] that the quaternion neural network reduces the complexity by about 30% compared with the real one under the same performance.

Meanwhile, the KAF has developed rapidly in recent years and can cope with nonlinear filtering problems using the kernel trick [8] of the reproducing kernel Hilbert space (RKHS) [9]. The corresponding algorithms were early studied in real- and complex-valued field, including the real kernel least-mean-square (KLMS) [10], real kernel recursive least-square (KRLS) [11], [12], [13], [14], [15] and real kernel recursive maximum correntropy [16], [17] and complex Gaussian KLMS algorithm [18]. Note that the real kernel filters need plentiful similar filters and ignore the connection between each component when dealing with 3-D or 4-D signals. The recent development of sensor technology makes 3-D or 4-D sensor signals accessible, and it requires proper algorithms, especially kernel algorithms to match the properties of hyper-dimension signals. When dealing with hyper-dimension signals, a quaternion kernel filter can combine the mutual information of four nonlinear constituents. Then the KAF was studied in quaternion-valued field using KLMS algorithm, such as quaternion stochastic information gradient algorithm [7] and quaternion minimum error entropy with fiducial point algorithm [19].

The derivation of quaternion algorithms, whether including a kernel or not, are based on the quaternion gradient. Different from the complex-valued and the real-valued domain, the quaternion domain is non-commutative. The quaternion gradient has undergone several versions. From 2009 to 2012, quaternion least-mean-squares (QLMS) [22], [23], HR-QLMS [24], and iQLMS [25], [26] were proposed. Later, the three types of gradient were proved to misuse the non-commutative property [27], [28], [29], [30]. From 2014 to 2019, the quaternion gradient was proposed using three approaches: the generalized Hamilton-real (GHR) calculus [27], [28], the quaternion product [29] and the quaternion involutions [30]. The above mentioned quaternion KAF algorithms in [7], [19] are based on the GHR calculus. The early quaternion RLS (QRLS) algorithms [20], [21] are based on the HR-QLMS [24], which prevents the generalization from RLS to KRLS using the kernel trick.

To the best of our knowledge, the quaternion KRLS has not been tackled by current methods. Generally speaking, the KRLS algorithm converges faster than the KLMS algorithm. This motivates us to study a KRLS algorithm in quaternion domain to coping with hyper-dimension nonlinear input signals. In this paper, we generalize the GHR calculus method [28] to derive a quaternion KRLS algorithm. Firstly, we investigate the QRLS algorithm; Secondly, we generalize the QRLS algorithm to the quaternion KRLS algorithm. The proposed KRLS algorithm can be applied in the nonlinear adaptive systems, such as biomedical signal processing and channel nonlinearities.

Section snippets

Quaternion algebra

A quaternion variable qH, consists of a real component and three imaginary components, written as a linear sumq=qa+iqb+jqc+kqdH,where qa,qb,qc,qdR. The conjugate of a quaternion isq*=qaıiqbjqckqdH.The relationship among the orthogonal unit vectors i,j,k areij=k,jk=i,ki=j,ijk=i2=j2=k2=1.The above three formulas define the quaterion algebra. The quaternion product is non-commutative, for example,i,j,k, instead ji=k. Given q1,q2H, we haveq1q2 ≠ q2q1.

Introducing the three perpendicular

Quaternion RLS

Similar with the above quaternion LMS algorithm, the quaternion RLS minimizes the cost function in estimating w. The cost function [14] is defined asJRLS=j=1ie*(j)e(j)+λwH(i)w(i),where λ is a real regularization parameter and e(i)=d(i)wHu(i). The gradient of Eq. (14) is obtained from (8), (12) and (13), yieldingJRLSw*=j=1i12u(j)e(j)*+λ2w(i)=j=1i12u(j)[d(j)w(i)Hu(j)]*+λ2w(i)=12j=1iu(j)[d*(j)uH(j)w(i)]+λ2w(i)=12j=1i[u(j)d*(j)u(j)uH(j)w(i)]+λ2w(i).Defining{R(i)=j=1iu(j)uH(j)+λI=R(i

Quaternion Kernel Method

The theory of the RKHS greatly promotes the development of the theory of adaptive signal processing. With the kernel trick, the KAF solves nonlinear filtering problems. In this section, we focus on providing the quaternion kernel RLS algorithm.

For a sequence of n input–output pairs, {(u1,d1),(u2,d2),,(un,dn)}, the input ui is transformed into the feature space F, which is an infinite-dimensional space. The parametric model of the feature space becomesy(i)=wH(i)φ(i),where φ(i)=ψ(ui)Hm, and ψ(ui

Quaternion RLS

We test the proposed QRLS algorithm and the real RLS algorithm. The linear model in quaternion model has been given in (9). The variables in (9) is expressed by{d=da+idb+jdc+kdd,woH=waoTiwboTjwcoTkwdoT,u=ua+iub+juc+kud,v=va+ivb+jvc+kvd,where (i) is omitted for simplicity.

The corresponding expression in the real domain is given by[dadbdcdd]=[uaTubTucTudTubTuaTudTucTucTudTuaTubTudTucTubTuaT][waowbowcowdo]+[vavbvcvd].Introduce{W=[waTwbTwcTwdT]T,UaT(i)=[uaT(i)ubT(i)ucT(i)udT(i)],UbT(i)=[ubT

Conclusion

We have generalized the GHR calculus method to calculate the quaternion RLS filter, and have proposed a new quaternion KRLS algorithm, which has the same form as the real-valued KRLS algorithm. The simulations verified the feasibility of the proposed algorithm. The proposed QRLS algorithm had almost the same steady error as that of the CFS under different SNRs. Moreover, the proposed quaternion KRLS converged faster than the algorithms, Quat-KLMS, and WL-Quat-KLMS.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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