An Application of Sinc Sum Function in Hilbert Transformer

Abstract

An application of the sinc sum function in Hilbert transformer (HT) is studied. The expression of the frequency re- sponse of HT is expressed with sinc sum functions. Some properties of sub-amplitude response of HT are proved by using the properties of the sinc sum function. A general HT formula is obtained theoretically and it contains a general window function. As an example three new window functions are obtained. Different from the existing window func- tions obtained from lowpass filters, these window functions are obtained directly from HT. Comparisons show that new windows are better than the Hanning, Hamming, Blackman and Kaiser windows in terms of HT performances.

Share and Cite:

Y. Wang, "An Application of Sinc Sum Function in Hilbert Transformer," Engineering, Vol. 5 No. 5A, 2013, pp. 1-8. doi: 10.4236/eng.2013.55A001.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. Feldman and S. Seibold, “Damage Diagnosis of Rotors: Application of Hilbert Transform and Multihypothesis Testing,” Journal of Vibration and Control, Vol. 5, 1999, pp. 421-442. doi:10.1177/107754639900500305
[2] W. J. Freeman, “Origin, Structure, and Role of Background EEG Activity. Part 3. Neural Frame Classification,” Clinical Neurophysiology, Vol. 116, No. 5, 2005, pp. 1118-1129. doi:10.1016/j.clinph.2004.12.023
[3] O. Deshmukh, C. Y. Espy-Wilson, A. Salomon and J. Singh, “Use of Temporal Information: Detection of Periodicity, Aperiodicity, and Pitch in Speech,” IEEE Transactions on Speech Audio Process, Vol. 13, No. 5, 2005, pp. 776-786. doi:10.1109/TSA.2005.851910
[4] F. L. Zarraga, “On-Line Extraction of Modal Characteristics from Power System Measurements Based on HilbertHuang Analysis,” International Conference on Electrical Engineering, Computing Science and Automatic Control, Toluca, 10-13 January 2009, pp. 1-6.
[5] C. J. Chun, Y. H. Lee, Y. G. Kim, H. K. Kim and C. S. Cho, “A Real-Time Audio Upmixing Method from Stereo to 7.1-Channel Audio,” Communication and Networking —Communications in Computer and Information Science, Vol. 120, 2010, pp. 162-171.
[6] F. Michael, “Hilbert Transform in Vibration Analysis,” Mechanical Systems and Signal Processing, Vol. 25, No. 3, 2011, pp. 735-802. doi:10.1016/j.ymssp.2010.07.018
[7] N. E. Huang, “Introduction to Hilbert-Huang Transform and Its Related Mathematical Problems,” In: N. E. Huang and S. S. P. Shen, Eds., Hilbert-Huang Transform and Its Applications, World Scientific, Singapore City, 2005, pp. 1-26.
[8] H. Luo, X. Fang and B. Ertas, “Hilbert Transform and Its Engineering Applications,” AIAA Journal, Vol. 47, 2009, pp. 923-932.
[9] H. W. Schüler and P. Steffen, “Halfband Filters and Hilbert Transformers,” Circuits, Systems and Signal Processing, Vol. 17, No. 2, 1998, pp. 137-164. doi:10.1007/BF01202851
[10] A. Hossen, “A New Fast Approximate Hilbert Transform with Different Applications,” IIUM Engineering Journal, Vol. 2, No. 2, 2001, pp. 21-27.
[11] S. W. A. Bergen and A. Antoniou, “Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function,” EURASIP Journal on Applied Signal Processing, Vol. 12, 2005, pp. 1910-1922. doi:10.1155/ASP.2005.1910
[12] H. Olkkonen, P. Pesola and J. T. Olkkonen, “Computation of Hilbert Transform via Discrete Cosine Transform,” Journal of Signal and Information Processing, Vol. 1, No. 1, 2010, pp. 18-23. doi:10.4236/jsip.2010.11002
[13] J. T. Olkkonen and H. Olkkonen, “Complex Hilbert Transform Filter,” Journal of Signal and Information Processing, Vol. 2, No. 2, 2011, pp. 112-116. doi:10.4236/jsip.2011.22015
[14] W. Yunlong, “Sinc Sum Function and Its Application on FIR Filter Design,” Acta Applicandae Mathematicae, Vol. 110, No. 3, 2010, pp. 1037-1056. doi:10.1007/s10440-009-9492-7
[15] Y. Wang, “An Effective Approach to Finding Differentiator Window Functions Based on Sinc Sum Function,” Circuits, Systems and Signal Processing, Vol. 31, No. 5, 2012, pp. 1809-1828. doi:10.1007/s00034-012-9399-9
[16] Y. Wang, “New Window Functions for the Design of Narrowband Lowpass Differentiators,” Circuits, Systems and Signal Processing, 2012. doi:10.1007/s00034-012-9536-5
[17] E. C. Ifeachor and B. W. Jervis, “Digital Signal Processing,” Publishing House of Electronics Industry, Beijing, 2003.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.