Presentation on theme: "An Introduction to HHT: Instantaneous Frequency, Trend, Degree of Nonlinearity and Non-stationarity Norden E. Huang Research Center for Adaptive Data Analysis."— Presentation transcript:
An Introduction to HHT: Instantaneous Frequency, Trend, Degree of Nonlinearity and Non-stationarity Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and Translational Medicine NCU, Zhongli, Taiwan, China
Outline Rather than the implementation details, I will talk about the physics of the method. What is frequency? How to quantify the degree of nonlinearity? How to define and determine trend? What is frequency? How to quantify the degree of nonlinearity? How to define and determine trend?
What is frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena.
Definition of Frequency Given the period of a wave as T ; the frequency is defined as
Other Definitions of Frequency : For any data from linear Processes
Definition of Power Spectral Density Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a wide- sense stationary random process:
Problem with Fourier Frequency Limited to linear stationary cases: same spectrum for white noise and delta function. Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost. Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum.
Surrogate Signal: Non-uniqueness signal vs. Power Spectrum I. Hello
The original data : Hello
The surrogate data : Hello
The Fourier Spectra : Hello
The Importance of Phase
To utilize the phase to define Instantaneous Frequency
Prevailing Views on Instantaneous Frequency The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer. J. Shekel, 1953 The uncertainty principle makes the concept of an Instantaneous Frequency impossible. K. Gröchennig, 2001
The Idea and the need of Instantaneous Frequency According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that Therefore, both wave number and frequency must have instantaneous values and differentiable.
Hilbert Transform : Definition
The Traditional View of the Hilbert Transform for Data Analysis
Traditional View a la Hahn (1995) : Data LOD
Traditional View a la Hahn (1995) : Hilbert
Traditional Approach a la Hahn (1995) : Phase Angle
Traditional Approach a la Hahn (1995) : Phase Angle Details
Traditional Approach a la Hahn (1995) : Frequency
The Real World Mathematics are well and good but nature keeps dragging us around by the nose. Albert Einstein
Why the traditional approach does not work?
Hilbert Transform a cos + b : Data
Hilbert Transform a cos + b : Phase Diagram
Hilbert Transform a cos + b : Phase Angle Details
Hilbert Transform a cos + b : Frequency
The Empirical Mode Decomposition Method and Hilbert Spectral Analysis Sifting ( Other alternatives, e.g., Nonlinear Matching Pursuit)
Empirical Mode Decomposition: Methodology : Test Data
Empirical Mode Decomposition: Methodology : data and m1
Empirical Mode Decomposition: Methodology : data & h1