Time-Varying Bandpass Filter Based on Assisted Signals for AM-FM Signal Separation: A Revisit

Guanlei Xu, Xiaotong Wang, Xiaogang Xu, Lijia Zhou, Limin Shao

Navgation Department of Dalian Naval Academy, Dalian, China.
Ocean Department of Dalian Naval Academy, Dalian, China.

DOI: 10.4236/jsip.2013.43031                

Abstract

In this paper, a new signal separation method mainly for AM-FM components blended in noises is revisited based on the new derived time-varying bandpass filter (TVBF), which can separate the AM-FM components whose frequencies have overlapped regions in Fourier transform domain and even have crossed points in time-frequency distribution (TFD) so that the proposed TVBF seems like a “soft-cutter” that cuts the frequency domain to snaky slices with rational physical sense. First, the Hilbert transform based decomposition is analyzed for the analysis of nonstationary signals. Based on the above analysis, a hypothesis under a certain condition that AM-FM components can be separated successfully based on Hilbert transform and the assisted signal is developed, which is supported by representative experiments and theoretical performance analyses on a error bound that is shown to be proportional to the product of frequency width and noise variance. The assisted signals are derived from the refined time-frequency distributions via image fusion and least squares optimization. Experiments on man-made and real-life data verify the efficiency of the proposed method and demonstrate the advantages over the other main methods.

Keywords

Time-Varying Bandpass Filter (TVBF); Hilbert Tranform; Assisted Signal; AM-FM Component; Time-Frequency Distribution (TFD)

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	X. D. Zhang, “Modern Signal Processing,” 2nd Edition, Tsinghua University Press, Beingjing, 2002.
[2]	G. Xu, X. Wang and X. Xu, “Time-Varying FrequencyShifting Signal Assisted Empirical Mode Decomposition Method for AM-FM Signals,” Mechanical Systems and Signal Processing, Vol. 23, No. 8, 2009, pp. 2458-2469. doi:10.1016/j.ymssp.2009.06.006
[3]	A. O. Boudraa and J. C. Cexus, “EMD-Based Signal Filtering,” IEEE Transactions on Instrumentation and Measurement, Vol. 56, No. 6, 2007, pp. 2196-2202. doi:10.1109/TIM.2007.907967
[4]	J. G. Proakis and D. G. Manolakis, “Digital Signal Processing: Principles, Algorithms, and Applications,” 3rd Edition, Prentice-Hall, Englewood Cliffs, 1996.
[5]	D. L. Donoho and I. M. Johnstone, “Ideal Spatial Adaptation via Wavelet Shrinkage,” Biometrica, Vol. 81, 1994, pp. 425-455. doi:10.1109/18.382009
[6]	D. L. Donoho, “De-Noising by Soft-Thresholding,” IEEE Transactions on Information Theory, Vol. 41, No. 3, 1995, pp. 613-627.
[7]	S. Mallat and Z. Zhang, “Matching Pursuits with TimeFrequency Dictionaries,” IEEE Transactions on Signal Processing, Vol. 41, No. 12, 1993, pp. 3397-3415. doi:10.1109/78.258082
[8]	G. L. Xu, X. T. Wang, X. G. Xu and L. J. Zhou, “Improved EMD for the Analysis of FM Signals,” Mechanical Systems and Signal Processing, Vol. 33, No. 11, 2012, pp. 181-196. doi:10.1016/j.ymssp.2012.07.003
[9]	F. Auger, P. Flandrin and P. Goncalves, “Time-Frequency in Action with Matlab,” 2000. http://www.researchgate.net/publication/50207277_Time-frequency_in_action_with_Matlab
[10]	B. Boashash, “Time-Frequency Signal Analysis and Processing—A Comprehensive Reference,” Elsevier Science, Oxford, 2003.
[11]	L. Cohen, “Time-Frequency Analysis,” Prentice-Hall, New York, 1995.
[12]	N. E. Huang, S. Zheng, R. L. Steven, et al., “The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear Non-Stationary Time Series Analysis,” Proceedings of the Royal Society A, Vol. 454, 1998, pp. 903-995. doi:10.1098/rspa.1998.0193
[13]	R. Deering and J. F. Kaiser, “The Use of a Masking Signal to Improve Empirical Mode Decomposition,” IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 4, 2005, pp. IV485-IV488.
[14]	N. Senroy, S. Suryanarayanan and P. F. Ribeiro, “An Improved Hilbert-Huang Method for Analysis of TimeVarying Waveforms in Power Quality,” IEEE Transactions on Power Systems, Vol. 22, No. 4, 2007, pp. 1843-1850. doi:10.1109/TPWRS.2007.907542
[15]	C. R. Gonzalez and E. R. Woods, “Digital Image Processing,” Pearson Education, 2nd Edition, Prentice Hall Press, Upper Saddle River, 2003.
[16]	G. Chen and Z. Wang, “A Signal Decomposition Theorem with Hilbert Transform and Its Application to Narrowband Time Series with Closely Spaced Frequency Components,” Mechanical Systems and Signal Processing, Vol. 28, 2012, pp. 258-279. doi:10.1016/j.ymssp.2011.02.002
[17]	R. Tao, B. Deng and Y. Wang, “Theory and Application of the Fractional Fourier Transform,” Tsinghua University Press, Beijing, 2009.
[18]	G. Rilling and P. Flandrin, “One or Two Frequencies? The Empirical Mode Decomposition Answers,” IEEE Transactions on Signal Processing, Vol. 56, No. 1, 2008, pp. 85-95. doi:10.1109/TSP.2007.906771
[19]	L. Angrisani, M. D’Arco, R. S. Lo Moriello, et al., “On the Use of the Warblet Transform for Instantaneous Frequency Estimation,” IEEE Transactions on Instrumentation and Measurement, Vol. 54, No. 4, 2005, pp. 1374-1380. doi:10.1109/TIM.2005.851060
[20]	J. P. Zhao and D. J. Huang, “Mirror Extending and Circular Spline Function for Empirical Mode Decomposition Method,” Journal of Zhejiang University Science, Vol. 2, No. 3, 2001, pp. 247-252. doi:10.1631/jzus.2001.0247
[21]	Y. J. Deng, et al., “An Approach for Ends Issue in EMD Method and Hilbert Transform,” Chinese Science Bulletin, Vol. 46, No. 3, 2001, pp. 903-1005.
[22]	J. Cheng, D. Yu and Y. Yang: “Discussion of the End Effects in Hilbert-Huang Transform,” Journal of Vibration and Shock, Vol. 24, No. 6, 2005, pp. 40-47.
[23]	J. Wang, Y. Peng and X. Peng, “Similarity Searching Based Boundary Effect Processing Method for Empirical Mode Decomposition,” Electronics Letters, Vol. 43, No. 1, 2007, pp. 1-2.
[24]	Y. Yang, W. M. Zhang, Z. K. Peng and G. Meng, “TimeFrequency Fusion Based on Polynomial Chirplet Transform for Non-Stationary Signals,” IEEE Transactions on Industrial Electronics, Vol. 60, No. 9, 2012, pp. 3948-3956. doi:10.1109/TIE.2012.2206331
[25]	E. Bedrosian, “A Product Theorem for Hilbert Transforms,” Memorandum RM-3439-PR, US Air Force Project RAND, 1962.