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Computing Instantaneous Frequency by normalizing Hilbert Transform

United States Patent

6,901,353
May 31, 2005
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This invention presents Normalized Amplitude Hilbert Transform (NAHT) and Normalized Hilbert Transform(NHT), both of which are new methods for computing Instantaneous Frequency. This method is designed specifically to circumvent the limitation set by the Bedorsian and Nuttal Theorems, and to provide a sharp local measure of error when the quadrature and the Hilbert Transform do not agree. Motivation for this method is that straightforward application of the Hilbert Transform followed by taking the derivative of the phase-angle as the Instantaneous Frequency (IF) leads to a common mistake made up to this date. In order to make the Hilbert Transform method work, the data has to obey certain restrictions.
Huang; Norden E. (Bethesda, MD)
The United States of America as represented by the Administrator of the National Aeronautics and Space Administration (Washington, DC)
10/ 615,365
July 8, 2003
ORIGIN OF INVENTION The inventor of the invention described herein is an employee of the United States Government. Therefore, the invention may be manufactured and used by or for the Government for governmental purposes without the payment of any royalties thereon or therefor. BACKGROUND OF THE INVENTION The term `Instantaneous Frequency (IF)` has elicited strong opinions amongst the data analysis community. The range covers from `banishing it forever from the dictionary of the communication engineer (Shekel, 1953)` to being a `conceptual innovation` in assigning physical significance to the nonlinearly distorted waveforms (Huang et al 1998)`. In between these extremes, there are plenty of more moderate opinions stressing the frustration of finding an acceptable definition. In general, most of the investigators accept the definition of classical wave theory, the derivative of the phase (see, for example, Cohen, 1995). But questions of its validity persist. Yet the need for instantaneous frequency is a real one for data from nonstationary and nonlinear processes. Certainly, the non-stationarity is a key feature here, but IF is even more important for nonlinear processes. If the process is non-stationary, the frequency should be ever changing, albeit at a slow rate. Then, there is a need for frequency value as a function of time, for the value will not be constant throughout. For the nonlinear cases, the oscillation can be viewed as intra-wave frequency modulation (Huang et al, 1998, 1999). Therefore, it is compelling to clarify the concept of, to settle these arguments on, and to provide a workable method for implementing the IF. To date the most popular and direct method to define Instantaneous Frequency is through the Hilbert Transform (HT). Yet practical difficulties of implementation make it not only useless, but also controversial. Straightforward application of HT and then taking the derivative of the phase-angle as the instantaneous frequency is the common mistake made up to this date (Hahn, 1995). In order to make the HT method work, the data has to obey certain restrictions. Some publications relating to Instantaneous Frequency are listed below: Bedrosian, E., 1963: On the quadrature approximation to the Hilbert Transform of modulated signals. Proc. IEEE, 51, 868-869. Boashash, B., 1992a: Estimating and interpreting the instantaneous frequency of a signal. Part I: Fundamentals, Proc. IEEE, 80, 520-538. Boashash, B., 1992a: Estimating and interpreting the instantaneous frequency of a signal. Part I: Algorithms and Applications, Proc. IEEE, 80, 540-568. Cohen, L., 1995: Time-frequency Analysis, Prentice Hall, Englewood Cliffs, N.J. Daubechies, I., 1992: Ten Lectures on Wavelets, Philadelphia SIAM. Flandrin, P., 1999: Time-Frequency/Time-Scale Analysis, Academic Press, San Diego, Calif. Gabor, D., 1946: Theory of communication, J. IEE, 93, 426-457. Goldstein, H., 1980: Classical Mechanics, 2.sup.nd Ed., Addison-Wesley, Reading, Mass. Hahn, S., 1995: Hilbert Transforms in Signal Processing, Artech House, Boston, Mass. Huang, et al. 1998: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. Soc. Lond., 454, 903-993. Huang, N. E., Z. Shen, R. S. Long, 1999: A New View of Nonlinear Water Waves--The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457. Kaiser, J. F., 1990: On Teager's energy algorithm and its generalization to continuous signals. Proc. 4.sup.th IEEE Signal Processing Workshop, Mohonk, N.Y. Landau, L. D. and E. M. Lifshitz, 1976: Mechanics, 3.sup.rd Ed. Pergamon, Oxford. Maragos, P., J. F. Kaiser, and T. F. Quatieri, 1993a: On amplitude and frequency demodulation using energy operators, IEEE Trans. Signal Processing, 41, 1532-1550. Maragos, P., J. F. Kaiser, and T. F. Quatieri, 1993b: Energy separation in signal modulation with application to speech analysis, IEEE Trans. Signal Processing, 41, 3024-3051. Meville, W. K., 1983: Wave modulation and breakdown, J. Fluid Mech., 128, 489-506. Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signals, Proceedings of IEEE, 54, 1458-1459. Rice, S. O., 1944a: Mathematical analysis of random noise, Bell Sys. Tech. Jl., 23, 282-310. Rice, S. O., 1944b: Mathematical analysis of random noise, III. Power spectrum and correlation functions, Bell Sys. Tech. Jl., 23, 310-332. Rice, S. O., 1945a: Mathematical analysis of random noise, III. Statistical properties of random noise currents, Bell Sys. Tech. Jl., 24, 46-108. Rice, S. O., 1944a: Mathematical analysis of random noise, IV. Noise through nonlinear devices, Bell Sys. Tech. Jl., 24, 109-156. Infeld, E. and G. Rowland, 1990: Nonlinear waves, solutons and chaos, Cambridge University Press, Cambridge. Shekel, J., 1953: Instantaneous Frequency. Proc. I.R.E. 41, 548. Van der Pol, B., 1946: The fundamental principles of frequency modulation, Proc. IEE, 93, 153-158. Whitham, G. B., 1975: Linear and Nonlinear Waves, New York, Wiley. The following U.S. Patents are incorporated by reference: U.S. Pat. No. 5,983,162 "Computer Implemented Empirical Mode Decomposition Method, Apparatus and Article of Manufacture" U.S. Pat. No. 6,381,559 "Empirical Mode Decomposition Apparatus, Method and Article of Manufacture for Analyzing Biological Signals and Performing Curve Fitting" U.S. Pat. No. 6,311,130 "Computer Implemented Empirical Mode Decomposition Method, Apparatus and Article of Manufacture for Two-Dimensional Signals"