AM  Vol.4 No.12 , December 2013
A Theoretical Foundation for the Widely Linear Processing of Quaternion-Valued Data
Author(s) Tohru Nitta
ABSTRACT

In this paper, we will give a theoretical foundation for a quaternion-valued widely linear estimation framework. The estimation error obtained with the quaternion-valued widely linear estimation method is proved to be smaller than that obtained using the usual quaternion-valued linear estimation method.


Cite this paper
Nitta, T. (2013) A Theoretical Foundation for the Widely Linear Processing of Quaternion-Valued Data. Applied Mathematics, 4, 1616-1620. doi: 10.4236/am.2013.412219.
References
[1]   B. Picinbono and P. Chevalier, “Widely Linear Estimation with Complex Data,” IEEE Transactions on Signal Processing, Vol. 43, No. 8, 1995, pp. 2030-2033.
http://dx.doi.org/10.1109/78.403373

[2]   H. Gerstacker, R. Schober and A. Lampe, “Receivers with Widely Linear Processing for Frequency-Selective Channels,” IEEE Transactions on Communications, Vol. 51, No. 9, 2003, pp. 1512-1523.
http://dx.doi.org/10.1109/TCOMM.2003.816992

[3]   R. Schober, W. H. Gerstacker and L. H.-J. Lampe, “DataAided and Blind Stochastic Gradient Algorithms for Widely Linear MMSE MAI Suppression for DS-CDMA,” IEEE Transactions on Signal Processing, Vol. 52, No. 3, 2004, pp. 746-756.
http://dx.doi.org/10.1109/TSP.2003.822359

[4]   D. P. Mandic and V. S. L. Goh, “Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models,” John Wiley and Sons Ltd., Hoboken, 2009.
http://dx.doi.org/10.1002/9780470742624

[5]   C. C. Took and D. P. Mandic, “The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes,” IEEE Transactions on Signal Processing, Vol. 57, No. 4, 2009, pp. 1316-1327.
http://dx.doi.org/10.1109/TSP.2008.2010600

[6]   T. Nitta, “A Quaternary Version of the Back-Propagation Algorithm,” Proceedings of the IEEE International Conference on Neural Networks (ICNN’95), Perth, 27 November-1 December, 1995, pp. 2753-2756.

[7]   P. Arena, L. Fortuna, G. Muscato and M. G. Xibilia, “Neural Networks in Multidimensional Domains,” Springer, London, 1998.

[8]   T. Isokawa, N. Matsui and H. Nishimura, “Quaternionic Neural Networks: Fundamental Properties and Applications,” In: T. Nitta, Ed., Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, Information Science Reference (IGI Global), Hershey, New York, 2009, pp. 411-439.
http://dx.doi.org/10.4018/978-1-60566-214-5.ch016

[9]   T. Nitta, “Widely Linear Processing of Hypercomplex Signals,” In: B.-L. Lu, L. Zhang and J. Kwok, Eds., Neural Information Processing, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2011, pp. 519-525.

[10]   C. C. Took and D. P. Mandic, “A Quaternion Widely Linear Adaptive Filter,” IEEE Transactions on Signal Processing, Vol. 58, No. 8, 2010, pp. 4427-4431.
http://dx.doi.org/10.1109/TSP.2010.2048323

[11]   J. Via, D. Ramirez and I. Santamaria, “Properness and Widely Linear Processing of Quaternion Random Vectors,” IEEE Transactions on Information Theory, Vol. 56, No. 7, 2010, pp. 3502-3515.
http://dx.doi.org/10.1109/TIT.2010.2048440

[12]   D. P. Mandic, C. Jahanchahi and C. C. Took, “A Quarternion Gradient Operator and its Applications,” IEEE Signal Processing Letters, Vol. 18, No. 1, 2011, pp. 47-50.
http://dx.doi.org/10.1109/LSP.2010.2091126

[13]   K. Gürlebeck, K. Habetha and W. SproBig, “Holomorphic Functions in the Plane and N-Dimensional Space,” Birkhauser, Basel, 2008.

 
 
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