Properties of a general quaternion-valued gradient operator and its applications to signal processing
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- 作者：Meng-di Jiang ; Yi Li ; Wei Liu
- 关键词：Quaternion ; Gradient operator ; Signal processing ; Least mean square (LMS) algorithm ; Nonlinear adaptive filtering ; Adaptive beamforming ; TN911.7 ; O29
- 刊名：Frontiers of Information Technology & Electronic Engineering
- 出版年：2016
- 出版时间：February 2016
- 年：2016
- 卷：17
- 期：2
- 页码：83-95
- 全文大小：464 KB
- 参考文献：Barthelemy, Q., Larue, A., Mars, J.I., 2014. About QLMS derivations. IEEE Signal Process. Lett., 21(2):240–243. http://dx.doi.org/10.1109/LSP.2014.2299066CrossRef MathSciNet
Brandwood, D.H., 1983. A complex gradient operator and its application in adaptive array theory. IEE Proc. F, 130(1):11–16. http://dx.doi.org/10.1049/ip-f-1.1983.0003MathSciNet
Compton, R.T., 1981. On the performance of a polarization sensitive adaptive array. IEEE Trans. Antenn. Propag., 29(5):718–725. http://dx.doi.org/10.1109/TAP.1981.1142651CrossRef
Ell, T.A., Sangwine, S.J., 2007. Quaternion involutions and anti-involutions. Comput. Math. Appl., 53(1):137–143. http://dx.doi.org/10.1016/j.camwa.2006.10.029CrossRef MathSciNet MATH
Ell, T.A., Le Bihan, N., Sangwine, S.J., 2014. Quaternion Fourier Transforms for Signal and Image Processing. Wiley, UK.CrossRef
Gentili, G., Struppa, D.C., 2007. A new theory of regular functions of a quaternionic variable. Adv. Math., 216(1):279–301. http://dx.doi.org/10.1016/j.aim.2007.05.010CrossRef MathSciNet MATH
Hawes, M., Liu, W., 2015. Design of fixed beamformers based on vector-sensor arrays. Int. J. Antenn. Propag., 2015:181937.1–181937.9. http://dx.doi.org/10.1155/2015/181937
Jiang, M.D., Li, Y., Liu, W., 2014a. Properties and applications of a restricted HR gradient operator. arXiv:1407.5178.
Jiang, M.D., Liu, W., Li, Y., 2014b. A general quaternionvalued gradient operator and its applications to computational fluid dynamics and adaptive beamforming. Proc. 19th Int. Conf. on Digital Signal Processing, p.821–826. http://dx.doi.org/10.1109/ICDSP.2014.6900781
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. http://dx.doi.org/10.1109/CSNDSP.2014.6923898
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. http://dx.doi.org/10.1016/j.sigpro.2004.04.001CrossRef MATH
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. http://dx.doi.org/10.1109/TSP.2007.896067CrossRef MathSciNet
Li, J., Compton, R.T., 1991. Angle and polarization estimation using ESPRIT with a polarization sensitive array. IEEE Trans. Antenn. Propag., 39:1376–1376. http://dx.doi.org/10.1109/8.99047CrossRef
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. http://dx.doi.org/10.1631/jzus.C1400038CrossRef
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.
Liu, W., Weiss, S., 2010. Wideband Beamforming: Concepts and Techniques.CrossRef
Wiley, UK. Mandic, D.P., Jahanchahi, C., Took, C.C., 2011. A quaternion gradient operator and its applications. IEEE Signal Process. Lett., 18(1):47–50. http://dx.doi.org/10.1109/LSP.2010.2091126CrossRef
Miron, S., Le Bihan, N., Mars, J.I., 2006. Quaternion-MUSIC for vector-sensor array processing. IEEE Trans. Signal Process., 54(4):1218–1229. http://dx.doi.org/10.1109/TSP.2006.870630CrossRef
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. http://dx.doi.org/10.1016/j.sigpro.2010.01.008CrossRef MATH
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. http://dx.doi.org/10.1109/83.760310CrossRef
Roberts, M.K., Jayabalan, R., 2015. An improved lowcomplexity sum-product decoding algorithm for lowdensity parity-check codes. Front. Inform. Technol. Electron. Eng., 16(6):511–518. http://dx.doi.org/10.1631/FITEE.1400269CrossRef
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.
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. http://dx.doi.org/10.1109/ICASSP.2015.7178713
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. http://dx.doi.org/10.1109/ICASSP.2014.6854840
Tao, J.W., 2013. Performance analysis for interference and noise canceller based on hypercomplex and spatiotemporal-polarisation processes. IET Radar Sonar Navig., 7(3):277–286. http://dx.doi.org/10.1049/iet-rsn.2012.0151CrossRef
Tao, J.W., Chang, W.X., 2014. Adaptive beamforming based on complex quaternion processes. Math. Prob. Eng., 2014:291249.1–291249.10. http://dx.doi.org/10.1155/2014/291249
Zhang, X.R., Liu, W., Xu, Y.G., et al., 2014. Quaternionvalued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint. Signal Process., 104:274–274. http://dx.doi.org/10.1016/j.sigpro.2014.04.006CrossRef - 作者单位：Meng-di Jiang (1)
Yi Li (2)
Wei Liu (1)
1. Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, S1 3JD, UK
2. School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK - 刊物类别：Computer Science, general; Electrical Engineering; Computer Hardware; Computer Systems Organization
- 刊物主题：Computer Science, general; Electrical Engineering; Computer Hardware; Computer Systems Organization and Communication Networks; Electronics and Microelectronics, Instrumentation; Communications Engine
- 出版者：Zhejiang University Press
- ISSN：2095-9230
文摘
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 quaternionvalued 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. Keywords Quaternion Gradient operator Signal processing Least mean square (LMS) algorithm Nonlinear adaptive filtering Adaptive beamforming