JSIP  Vol.3 No.4 , November 2012
Comparing the Time-Deformation Method with the Fractional Fourier Transform in Filtering Non-Stationary Processes
Abstract: The classical linear filter is able to extract components from multi-component stochastic processes where the frequencies of components do not overlap over time, but fail for those processes where the frequencies overlap over time. In this paper, we discuss two filtering methods for non-stationary processes: the G-filtering method and the Fractional Fourier transform (FrFT) method. The FrFT method is mainly designed for linear chirp signals where the frequency is linearly changing with time. The G-filter can be used to filter signals with wide range of frequency behaviors such as linear chirps, quadratic chirps and other type of chirp signals with strong time-varying frequency behavior. If frequencies of the components can be approximated or separated by a straight line or a polynomial curve, the G-filter can successfully extract components from the original series. We show that the G-filter is applicable to a wider variety of filtering applications than methods such as the FrFT which require data of a specified frequency behavior.
Cite this paper: M. Xu, W. A. Woodward and H. L. Gray, "Comparing the Time-Deformation Method with the Fractional Fourier Transform in Filtering Non-Stationary Processes," Journal of Signal and Information Processing, Vol. 3 No. 4, 2012, pp. 491-501. doi: 10.4236/jsip.2012.34062.

[1]   A. Papoulis, “Probability, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 1965.

[2]   R. H. Shumway and D. S. Stoffer, “Time Series Analysis and Its Applications,” Springer, New York, 2000.

[3]   S. Butterworth, “On the Theory of Filter Amplifiers,” Experimental Wireless and the Wireless Engineer, Vol. 7, 1930, pp. 536-541.

[4]   M. Xu, K. B. Cohlmia, W. A. Woodward and H. L. Gray, “G-Filtering Nonstationary Time Series,” Journal of Probability and Statistics, Vol. 2012, 2012, Article ID: 738636. doi:10.1155/2012/738636

[5]   H. L. Gray, C. C. Vijverberg and W. A. Woodward, “Non-stationary Data Analysis by Time Deformation,” Communication in Statistics-Theory and Methods, Vol. 34, No. 1, 2005, pp. 163-192. doi:10.1081/STA-200045869

[6]   H. Jiang, H. L. Gray and W. A. Woodward, “Time- Frequency Analysis—G(λ) stationary Processes,” Computational Statistics & Data Analysis, Vol. 51, No. 3, 2006, pp. 1997-2028. doi:10.1016/j.csda.2005.12.011

[7]   L. Liu, “Spectral Analysis with Time-Varying Frequency,” Ph.D. Dissertation, Southern Methodist University, Boaz Lane Dallas, 2004. doi:10.1016/j.jspi.2010.04.033

[8]   S. D. Robertson, H. L. Gray and W. A. Woodward, “The Generalized Linear Chirp Process,” Journal of Statistical Planning and Inference, Vol. 140, No. 12, 2010, pp. 3676-3687.

[9]   B. Boashash, “Time Frequency Analysis,” Elsevier, Oxford, 2003.

[10]   R. G. Baraniuk and D. L. Jones, “Unitary Equivalence: A New Twist on Signal Processing,” IEEE Transactions on Signal Processing, Vol. 43, No. 11, 1995, pp. 2269-2282. doi:10.1109/78.469861

[11]   M. Xu, “Filtering Non-Stationary Time Series by Time Deformation,” Ph.D. Dissertation, Southern Methodist University, Dallas, 2004.

[12]   H. M. Ozaktas, O. Arikan, M. A. Kutay and G. Bozdagt, “Digital Computation of the Fractional Fourier Transform,” IEEE Transactions on Signal Processing, Vol. 44, No. 9, 1996, pp. 2141-2150. doi:10.1109/78.536672

[13]   C. Capus and K. Brown, “Short-Time Fractional Fourier Methods for the Time-Frequency Representation of Chirp Signals,” Journal of Acoustical Society of America, Vol. 113, No. 6, 2003, pp. 3253-3263. doi:10.1121/1.1570434

[14]   H. M. Ozaktas, Z. Zalevsky and M. A. Kutay, “The Fractional Fourier Transform,” John Willey & Sons, Ltd., New York, 2000.