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Computation of Hilbert Transform via Discrete Cosine Transform

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DOI: 10.4236/jsip.2010.11002    7,072 Downloads   13,985 Views   Citations

ABSTRACT

Hilbert transform (HT) is an important tool in constructing analytic signals for various purposes, such as envelope and instantaneous frequency analysis, amplitude modulation, shift invariant wavelet analysis and Hilbert-Huang decomposition. In this work we introduce a method for computation of HT based on the discrete cosine transform (DCT). We show that the Hilbert transformed signal can be obtained by replacing the cosine kernel in inverse DCT by the sine kernel. We describe a FFT-based method for the computation of HT and the analytic signal. We show the usefulness of the proposed method in mechanical vibration and ultrasonic echo and transmission measurements.

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H. Olkkonen, P. Pesola and J. 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.

References

[1] N. E. Huang, Z. Shen, and S. R. Long, M. C. Wu, E. H. Shih, Q. Zheng, C. C. Tung, and H. H. Liu, “The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Nonstationary Time Series Analysis,” Proceedings of the Royal Society of London A, Vol. 454, pp. 903-995, 1998.
[2] I. W. Selesnick, “Hilbert Transform Pairs of Wavelet Bases,” IEEE Signal Processing Letters, Vol. 8, No. 6, June 2001, pp. 170-173.
[3] H. Ozkaramanli and R. Yu, “On the Phase Condition and Its Solution for Hilbert Transform Pairs of Wavelet Bases,” IEEE Transactions on Signal Processing, Vol. 51, No. 12, December 2003, pp. 3293-3294.
[4] R. Yu and H. Ozkaramanli, “Hilbert Transform Pairs of Biorthogonal Wavelet Bases,” IEEE Transactions on Signal Processing, Vol. 54, No. 6, part 1, June 2006, pp. 2119-2125.
[5] H. Olkkonen, J. T. Olkkonen and P. Pesola, “FFT Based Computation of Shift Invariant Analytic Wavelet Transform,” IEEE Signal Processing Letters, Vol. 14, No. 3, March 2007, pp. 177-180.
[6] W. M. Moon, A. Ushah and B. Bruce, “Application of 2-D Hilbert Transform in Geophysical Imaging with Potential Field Data,” IEEE Transactions on Geoscience and Remote Sensing, Vol. 26, No. 5, September 1988, pp. 502-510.
[7] H. Olkkonen, P. Pesola, J. T. Olkkonen and H. Zhou, “Hilbert Transform Assisted Complex Wavelet Trans- form for Neuroelectric Signal Analysis,” Journal of Neuroscience Methods, Vol. 151, No. 2, pp. 106-113, 2006.
[8] H. Kanai, Y. Koiwa and J. Zhang, “Real-Time Measurements of Local Myocardium Motion and Arterial Wall Thickening,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 46, No. 5, September 1999, pp. 1229-1241.
[9] S. M. Shors, A. V. Sahakian, H. J. Sih and S. Swiryn, “A Method for Determining High-Resolution Activation Time Delays in Unipolar Cardiac Mapping,” IEEE Tran- sactions on Biomedical Engineering, Vol. 43, No. 12, De- cember 1996, pp. 1192-1196.
[10] A. K. Barros and N. Ohnishi, “Heart Instantaneous Frequency (HIF): An Alternative Approach to Extract Heart Rate Variability,” IEEE Transactions on Biomedical Engineering, Vol. 48, No. 8, August 2001, pp. 850-855.
[11] O. W. Kwon and T. W. Lee, “Phoneme Recognition Using ICA-Based Feature Extraction and Transformation,” Signal Processing, Vol. 84, No. 6, 2004, pp. 1005-1019.
[12] A. V. Oppenheim and R. W. Schafer, “Discrete-Time Signal Processing,” Prentice-Hall, Englewood Cliffs, 1989.
[13] K. L. Peacock, “Kaiser-Bessel Weighting of the Hilbert Transform High-Cut Filter,” IEEE Transactions on Acous?tics, Speech, and Signal Analysis, Vol. 33, No. 1, February 1985, pp. 329-331.
[14] A. Rao and R. Kumaresan, “A Parametric Modeling Approach to Hilbert Transformation,” IEEE Signal Processing Letters, Vol. 5, No. 1, January 1998, pp. 15-17.
[15] R. Kumaresan, “An Inverse Signal Approach to Com- puting the Envelope of a Real Valued Signal,” IEEE Signal Processing Letters, Vol. 5, No. 10, October 1998, pp. 256-259.
[16] S. C. Pei and S. B. Jaw, “Computation of Discrete Hilbert Transform through Fast Hartley Transform,” IEEE Tran- sactions on Circuits and Systems, Vol. 36, No. 9, 1989, pp. 1251-1252.
[17] N. Ahmed, T. Natajaran and K. R. Rao, “Discrete Cosine Transform,” IEEE Transactions on Computers, Vol. 23, 1974, pp. 90-94.
[18] C. W. Kok, “Fast Algorithm for Computing Discrete Cosine Transform,” IEEE Transactions on Signal Processing, Vol. 45, No. 3, March 1997, pp. 757-760.
[19] G. Bi and L. W. Yu, “DCT Algorithms for Composite Sequence Lengths,” IEEE Transactions on Signal Processing, Vol. 46, No. 3, March 1998, pp. 554-562.
[20] V. Britanak, P. Yip and K. R. Rao, Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations, Academic Press Inc., Elsevier Science, Amsterdam, 2007
[21] H. Malvar, “Fast Computation of the Discrete Cosine Transform and the Discrete Hartley Transform,” IEEE Transactions on Acoustics, Speech, Signal Processing, Vol. 35, No. 10, October 1987, pp. 1484-1485.

  
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