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

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 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

References

[1] 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.2014.2299066

[2] 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-1.1983.0003

[3] 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.1981.1142651

[4] 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.2006.10.029

[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. http://dx.doi.org/10.1016/j.aim.2007.05.010

[7] 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

[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 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

[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. http://dx.doi.org/10.1109/CSNDSP.2014.6923898

[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. http://dx.doi.org/10.1016/j.sigpro.2004.04.001

[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. http://dx.doi.org/10.1109/TSP.2007.896067

[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. http://dx.doi.org/10.1109/8.99047

[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. http://dx.doi.org/10.1631/jzus.C1400038

[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. http://dx.doi.org/10.1109/LSP.2010.2091126

[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. http://dx.doi.org/10.1109/TSP.2006.870630

[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. http://dx.doi.org/10.1016/j.sigpro.2010.01.008

[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. http://dx.doi.org/10.1109/83.760310

[21] 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.1400269

[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. http://dx.doi.org/10.1109/ICASSP.2015.7178713

[24] Talebi, S.P., Xu, D.P., Kuh, A., , 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

[25] 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.2012.0151

[26] 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

[27] Zhang, X.R., Liu, W., Xu, Y.G., , 2014. Quaternionvalued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint. Signal Process., 104:274–283. http://dx.doi.org/10.1016/j.sigpro.2014.04.006