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Signal Processing 178 (2021) 107810
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Signal Processing
journal homepage: www.elsevier.com/locate/sigpro
Short communication
Quaternion kernel recursive least-squares algorithm
Gang Wang
a
, Jingci Qiao
a
, Rui Xue
b , ∗
, Bei Peng
c
a
Center for Robotics, School of Information and Communication Engineering, University of Electronic Science and Technology of China,
Chengdu 61 1 731, PR China
b
School of Electronics and Information Engineering, Beihang University, Beijing, PR China
c
School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 61 1 731, PR China
a r t i c l e i n f o
Article history:
Received 7 January 2020
Accepted 8 September 2020
Available online 10 September 2020
Keywords:
Quaternion involutions
Quaternion kernel adaptive filter
Recursive least squares
Kernel recursive least square
a b s t r a c t
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 conver-
gence, and simulations are described demonstrating the performance of the proposed algorithm.
©2020 Elsevier B.V. All rights reserved.
1.
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 track-
ing [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 infor-
mation 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 cor-
responding 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) [11–15] and real ker-
nel 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
∗
Corresponding author.
E-mail addresses: wanggang_hld@uestc.edu.cn (G. Wang), qiaojingci@126.com (J.
Qiao), xuerui@buaa.edu.cn (R. Xue), beipeng@uestc.edu.cn (B. Peng).
when dealing with 3-D or 4-D signals. The recent development
of sensor technology makes 3-D or 4-D sensor signals accessi-
ble, and it requires proper algorithms, especially kernel algorithms
to match the properties of hyper-dimension signals. When deal-
ing 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 al-
gorithm [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 under-
gone 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 mis-
use the non-commutative property [27–30] . 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 men-
tioned quaternion KAF algorithms in [7,19] are based on the GHR
calculus. The early quaternion RL S (QRL S) 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 algo-
rithm converges faster than the KLMS algorithm. This motivates us
https://doi.org/10.1016/j.sigpro.2020.107810
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