Full Text:   <1702>

Summary:  <1509>

CLC number: TN911.7; O29

On-line Access: 2016-02-02

Received: 2015-10-15

Revision Accepted: 2016-01-06

Crosschecked: 2016-01-14

Cited: 4

Clicked: 9488

Citations:  Bibtex RefMan EndNote GB/T7714

 ORCID:

Wei Liu

http://orcid.org/0000-0003-2968-2888

-   Go to

Article info.
Open peer comments

Frontiers of Information Technology & Electronic Engineering  2016 Vol.17 No.2 P.83-95

http://doi.org/10.1631/FITEE.1500334


Properties of a general quaternion-valued gradient operator and its applications to signal processing


Author(s):  Meng-di Jiang, Yi Li, Wei Liu

Affiliation(s):  1Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK; more

Corresponding email(s):   w.liu@sheffield.ac.uk

Key Words:  Quaternion, Gradient operator, Signal processing, Least mean square (LMS) algorithm, Nonlinear adaptive filtering, Adaptive beamforming


Share this article to: More |Next Article >>>

Meng-di Jiang, Yi Li, Wei Liu. Properties of a general quaternion-valued gradient operator and its applications to signal processing[J]. Frontiers of Information Technology & Electronic Engineering, 2016, 17(2): 83-95.

@article{title="Properties of a general quaternion-valued gradient operator and its applications to signal processing",
author="Meng-di Jiang, Yi Li, Wei Liu",
journal="Frontiers of Information Technology & Electronic Engineering",
volume="17",
number="2",
pages="83-95",
year="2016",
publisher="Zhejiang University Press & Springer",
doi="10.1631/FITEE.1500334"
}

%0 Journal Article
%T Properties of a general quaternion-valued gradient operator and its applications to signal processing
%A Meng-di Jiang
%A Yi Li
%A Wei Liu
%J Frontiers of Information Technology & Electronic Engineering
%V 17
%N 2
%P 83-95
%@ 2095-9184
%D 2016
%I Zhejiang University Press & Springer
%DOI 10.1631/FITEE.1500334

TY - JOUR
T1 - Properties of a general quaternion-valued gradient operator and its applications to signal processing
A1 - Meng-di Jiang
A1 - Yi Li
A1 - Wei Liu
J0 - Frontiers of Information Technology & Electronic Engineering
VL - 17
IS - 2
SP - 83
EP - 95
%@ 2095-9184
Y1 - 2016
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/FITEE.1500334


Abstract: 
The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued quaternion functions and linear quaternion-valued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator, which comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in the real domain. As an application, the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided. Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm.

一般四元数函数梯度的定义、特性及在信号处理领域的应用

目的:随着四元数在信号处理各个领域越来越广泛的应用,基于四元数的信号处理理论也获得了快速发展。然而,制约其进一步应用的一个瓶颈就是对一般四元数函数的梯度的定义及特性还缺乏清晰并有说服力的描述。本文就试图对这一问题进行探索。
创新点:在信号处理中,虽然很多优化函数的值都是实数,但在进行优化时,尤其是在非线性信号处理中,经常会遇到对取值为四元数的四元数函数求梯度。不同于以往只适用于实数值四元数函数梯度的定义,本文第一次就一般四元数函数的梯度给出了一个自洽的定义,并对其特性进行了详细的研究和描述。基于以上研究,本文对四元数值的最小均方(LMS)自适应算法,以及一个有代表性的非线性自适应算法进行了推导,并以矢量传感器阵列波束形成为例进行了计算机模拟。

关键词:四元数;梯度;信号处理;最小均方算法;非线性自适应滤波;波束形成

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1]Barthelemy, Q., Larue, A., Mars, J.I., 2014. About QLMS derivations. IEEE Signal Process. Lett., 21(2):240-243.

[2]Brandwood, D.H., 1983. A complex gradient operator and its application in adaptive array theory. IEE Proc. F, 130(1):11-16.

[3]Compton, R.T., 1981. On the performance of a polarization sensitive adaptive array. IEEE Trans. Antenn. Propag., 29(5):718-725.

[4]Ell, T.A., Sangwine, S.J., 2007. Quaternion involutions and anti-involutions. Comput. Math. Appl., 53(1):137-143.

[5]Ell, T.A., Le Bihan, N., Sangwine, S.J., 2014. Quaternion Fourier Transforms for Signal and Image Processing. Wiley, UK.

[6]Gentili, G., Struppa, D.C., 2007. A new theory of regular functions of a quaternionic variable. Adv. Math., 216(1):279-301.

[7]Hawes, M., Liu, W., 2015. Design of fixed beamformers based on vector-sensor arrays. Int. J. Antenn. Propag., 2015:181937.1-181937.9.

[8]Jiang, M.D., Li, Y., Liu, W., 2014a. Properties and applications of a restricted HR gradient operator. arXiv:1407.5178.

[9]Jiang, M.D., Liu, W., Li, Y., 2014b. A general quaternion-valued gradient operator and its applications to computational fluid dynamics and adaptive beamforming. Proc. 19th Int. Conf. on Digital Signal Processing, p.821-826.

[10]Jiang, M.D., Liu, W., Li, Y., 2014c. A zero-attracting quaternion-valued least mean square algorithm for sparse system identification. Proc. 9th Int. Symp. on Communication Systems, Networks and Digital Signal Processing, p.596-599.

[11]Le Bihan, N., Mars, J., 2004. Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Signal Process., 84(7):1177-1199.

[12]Le Bihan, N., Miron, S., Mars, J.I., 2007. MUSIC algorithm for vector-sensors array using biquaternions. IEEE Trans. Signal Process., 55(9):4523-4533.

[13]Li, J., Compton, R.T., 1991. Angle and polarization estimation using ESPRIT with a polarization sensitive array. IEEE Trans. Antenn. Propag., 39:1376-1383.

[14]Liu, H., Zhou, Y.L., Gu, Z.P., 2014. Inertial measurement unit-camera calibration based on incomplete inertial sensor information. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 15(11):999-1008.

[15]Liu, W., 2014. Antenna array signal processing for a quaternion-valued wireless communication system. Proc. IEEE Benjamin Franklin Symp. on Microwave and Antenna Sub-systems, arXiv:1504.02921.

[16]Liu, W., Weiss, S., 2010. Wideband Beamforming: Concepts and Techniques. Wiley, UK.

[17]Mandic, D.P., Jahanchahi, C., Took, C.C., 2011. A quaternion gradient operator and its applications. IEEE Signal Process. Lett., 18(1):47-50.

[18]Miron, S., Le Bihan, N., Mars, J.I., 2006. Quaternion-MUSIC for vector-sensor array processing. IEEE Trans. Signal Process., 54(4):1218-1229.

[19]Parfieniuk, M., Petrovsky, A., 2010. Inherently lossless structures for eight- and six-channel linear-phase paraunitary filter banks based on quaternion multipliers. Signal Process., 90(6):1755-1767.

[20]Pei, S.C., Cheng, C.M., 1999. Color image processing by using binary quaternion-moment-preserving thresholding technique. IEEE Trans. Image Process., 8(5):614-628.

[21]Roberts, M.K., Jayabalan, R., 2015. An improved low-complexity sum-product decoding algorithm for low-density parity-check codes. Front. Inform. Technol. Electron. Eng., 16(6):511-518.

[22]Sangwine, S.J., Ell, T.A., 2000. The discrete Fourier transform of a colour image. Proc. Image Processing II: Mathematical Methods, Algorithms and Applications, p.430-441.

[23]Talebi, S.P., Mandic, D.P., 2015. A quaternion frequency estimator for three-phase power systems. Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, p.3956-3960.

[24]Talebi, S.P., Xu, D.P., Kuh, A., et al., 2014. A quaternion least mean phase adaptive estimator. Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, p.6419-6423.

[25]Tao, J.W., 2013. Performance analysis for interference and noise canceller based on hypercomplex and spatio-temporal-polarisation processes. IET Radar Sonar Navig., 7(3):277-286.

[26]Tao, J.W., Chang, W.X., 2014. Adaptive beamforming based on complex quaternion processes. Math. Prob. Eng., 2014:291249.1-291249.10.

[27]Zhang, X.R., Liu, W., Xu, Y.G., et al., 2014. Quaternion-valued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint. Signal Process., 104:274-283.

Open peer comments: Debate/Discuss/Question/Opinion

<1>

Please provide your name, email address and a comment





Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2022 Journal of Zhejiang University-SCIENCE