Complex Hilbert Transform Filter

Juuso T. Olkkonen, Hannu Olkkonen

.

DOI: 10.4236/jsip.2011.22015                

Abstract

Hilbert transform is a basic tool in constructing analytical signals for a various applications such as amplitude modulation, envelope and instantaneous frequency analysis, quadrature decoding, shift-invariant multi-rate signal processing and Hilbert-Huang decomposition. This work introduces a complex Hilbert transform (CHT) filter, where the real and imaginary parts are a Hilbert transform pair. The CHT filtered signal is analytic, i.e. its Fourier transform is zero in negative frequency range. The CHT filter is constructed by half-sample delay operators based on the B-spline transform interpolation and decimation procedure. The CHT filter has an ideal phase response and the magnitude response is maximally flat in the frequency range 0 ≤ ω ≤ π. The CHT filter has integer coefficients and the implementation in VLSI requires only summations and register shifts. We demonstrate the feasibility of the CHT filter in reconstruction of the sign modulated CMOS logic pulses in a fibre optic link.

Keywords

Hilbert Transform, Analytic Signal, Fractional Delay Filters

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. E. Huang, Z. Shen, 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,” Pro- ceedings of the Royal Society of London A, Vol. 454, No. 1971, 1998, pp. 903-995.
[2]	I. W. Selesnick, “Hilbert Transform Pairs of Wavelet Bases,” IEEE Signal Processing Letters, Vol. 8, No. 6, 2001, pp. 170-173. doi:10.1109/97.923042
[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, 2003, pp. 3293-3294. doi:10.1109/TSP.2003.818996
[4]	R. Yu and H. Ozkaramanli, “Hilbert Transform Pairs of Biorthogonal Wavelet Bases,” IEEE Transactions on Signal Processing, Vol. 54, No. 6, 2006, pp. 2119-2125. doi:10.1109/TSP.2006.874293
[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, 2007, pp. 177-180. doi:10.1109/LSP.2006.879983
[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, 1988, pp. 502-510. doi:10.1109/36.7674
[7]	H. Olkkonen, P. Pesola, J. T. Olkkonen and H. Zhou, “Hilbert Transform Assisted Complex Wavelet Transform for Neuroelectric Signal Analysis,” Journal of Neuroscience Methods, Vol. 151, No. 2, 2006, pp. 106-113. doi:10.1016/j.jneumeth.2005.06.028
[8]	H. Kanai, Y. Koiwa and J. Zhang, “Real-Time Measurements of Local Myocardium Motion and Arterial Wall Thickening,” IEEE Transaction on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 46, No. 5, 1999, pp. 1229-1241. doi:10.1109/58.796128
[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- saction on Biomedical Engineering, Vol. 43, No. 12, 1996, pp. 1192-1196.
[10]	A. K. Barros and N. Ohnishi, “Heart Instantaneous Frequency (HIF): An Alternative Approach to Extract Heart Rate Variability,” IEEE Transaction on Biomedical Engineering, Vol. 48, No. 8, 2001, pp. 850-855. doi:10.1109/10.936361
[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. doi:10.1016/j.sigpro.2004.03.004
[12]	A. V. Oppenheim and R. W. Schafer, “Discrete-Time Signal Processing,” Prentice-Hall, Englewood Cliffs, NJ, 1989.
[13]	A. Rao and R. Kumaresan, “A Parametric Modeling Approach to Hilbert Transformation,” IEEE Signal Proces- sing Letters, Vol. 5, No. 1, 1998, pp. 15-17. doi:10.1109/97.654868
[14]	R. Kumaresan, “An Inverse Signal Approach to Computing the Envelope of a Real Valued Signal,” IEEE Signal Processing Letters, Vol. 5, No. 10, 1998, pp. 256-259. doi:10.1109/97.720558
[15]	K. L. Peacock, “Kaiser-Bessel Weighting of the Hilbert Transform High-Cut Filter,” IEEE Transaction on Acoustics, Speech, and Signal Analysis, Vol. 33, No. 1, 1985, pp. 329-331. doi:10.1109/TASSP.1985.1164514
[16]	J. T. Olkkonen and H. Olkkonen, “Fractional Delay Filter Based on the B-Spline Transform,” IEEE Signal Proces- sing Letters, Vol. 14, No. 2, 2007, pp. 97-100. doi:10.1109/LSP.2006.882103
[17]	T. Laakso, V. Valimaki, M. Karjalainen and U. K. Laine, “Splitting the Unit Delay. Tools for Fractional Delay Filter Design,” IEEE Signal Processing Magazine, Vol. 13, No. 1, 1996, pp. 30-60. doi:10.1109/79.482137
[18]	S. C. Pei and C. C. Tseng, “An Efficient Design of a Variable Fractional Delay Filter Using a First-Order Differentiator,” IEEE Signal Processing Letters, Vol. 10, No. 10, 2003, pp. 307-310. doi:10.1109/LSP.2003.815616
[19]	S. C. Pei and P. H. Wang, “Closed-Form Design of All-Pass Fractional Delay,” IEEE Signal Processing Letters, Vol. 11, No. 10, 2004, pp. 788-791. doi:10.1109/LSP.2004.835473
[20]	H. Johansson and P. Lowenborg, “Reconstruction of Nonuniformly Sampled Bandlimited Signals by Means of Digital Fractional Delay Filters,” IEEE Transaction on Signal Processing, Vol. 50, No. 11, 2002, pp. 2757-2767. doi:10.1109/TSP.2002.804089
[21]	C. C. Tseng, “Digital Integrator Design Using Simpson Rule and Fractional Delay Filter,” IEE Proceedings of Vision, Image and Signal Processing, Vol. 153, No. 1, 2006, pp. 79-85. doi:10.1049/ip-vis:20045208
[22]	M. Unser, A. Aldroubi and M. Eden, “Polynomial Spline Approximations: Filter Design and Asymptotic Equivalence with Shannon’s Sampling Theorem,” IEEE Transactions on Information Theory, Vol. 38, No. 1, 1992, pp. 95-103. doi:10.1109/18.108253