On the Hilbert-Huang Transform Data Processing System Development
Semion Kizhner1, Thomas P. Flatley1, Dr. Norden E. Huang1,
Karin Blank1, Evette Conwell1 and Darrell Smith2
1NASA/ Goddard Space Flight Center Greenbelt MD, 20771
2Orbital Sciences Corporation
One of the main heritage tools used in scientific and engineering data spectrum analysis is the Fourier Integral Transform and its high performance digital equivalent – the Fast Fourier Transform (FFT). The Fourier view of nonlinear mechanics that had existed for a long time, and the associated FFT (fairly recent development), carry strong a-priori assumptions about the source data, such as linearity and of being stationary. Natural phenomena measurements are essentially nonlinear and nonstationary. A very recent development at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC), known as the Hilbert-Huang Transform (HHT) proposes a novel approach to the solution for the nonlinear class of spectrum analysis problems. Using the Empirical Mode Decomposition (EMD) followed by the Hilbert Transform of the empirical decomposition data (HT) , , , , the HHT allows spectrum analysis of nonlinear and nonstationary data by using an engineering a-posteriori data processing, based on the EMD algorithm. This results in a non-constrained decomposition of a source real value data vector into a finite set of Intrinsic Mode Functions (IMF) that can be further analyzed for spectrum interpretation by the classical Hilbert Transform. This paper describes phase one of the development of a new engineering tool, the HHT Data Processing System (HHTDPS). The HHTDPS allows applying the HHT to a data vector in a fashion similar to the heritage FFT. It is a generic, low cost, high performance personal computer (PC) based system that implements the HHT computational algorithms in a user friendly, file driven environment. This paper also presents a quantitative analysis for a complex waveform data sample, a summary of technology commercialization efforts and the lessons learned from this new technology development.
Oscillatory phenomena are omnipresent and the desire to learn about the oscillatory behavior of a signal is both natural and practically useful. Signal characteristics, the study of a signal by its decomposition into simpler components, the heritage signal processing method for linear systems and data, and the novel method that is applicable to non-linear and non-stationary data are presented in this introduction.
The characteristic parameters of an oscillatory signal are period T>0, oscillation fundamental frequency f0, amplitude a, phase (t), phase shift C within (t), polarization, as well
as signal average value, energy and power. These are described in detail in Section 1.5.
When dealing with a signal that is not susceptible to a brute force study of its oscillatory behavior, it is tempting to decompose the signal into simpler oscillatory behavior components (analysis) and in such a way that allows straightforward reconstruction of the source signal from the decomposition components (synthesis). The signal synthesis from a subset of components yields the source signal approximation and always related to an approximation residue affects. The decomposition, in turn, can use some set of basic functions (basis) for component representation, such as polynomials or trigonometric sine and cosine functions, or periodic exponential functions of a complex variable. Any complete set of functions can be used to represent arbitrary functions. There are also multitudes of ways to decompose anything, be to a simple signal with a constant amplitude or a complex non-linear function. For example, a signal with a constant amplitude in time x(t)=3/2 can be represented as a sum of squared trigonometric cosine functions of time (components), namely
3/2 = x(t) = cos2(t - a) + cos2(t) + cos2(t + a)
for a = 2*pi/3 and all t (in radians).
However, there is very little knowledge about this signal oscillatory behavior (or absence of it) available from such arbitrary decomposition into non-linear cosine-based components.
The heritage spectrum analysis method is based on the Fourier theoretics, a linear decomposition that is especially convenient when a signal originates within a linear and stationary oscillatory process. The Fourier series for periodic signals with a finite period T>0 or the Fourier Integral Transform (also called the Fourier Transform) for spectrum analysis of non-periodic linear and stationary functions is implemented for signals that satisfy the Dirichlet criteria which is described in Section 1.5. A signal with period T=0 is a signal with a constant amplitude in time. It has no nonzero frequency oscillatory components and is also fully covered by the Fourier series. This linear decomposition into sine and cosine components, employed by Fourier series and the Fourier Transform, is of obvious advantage since it provides a direct solution for a linear and stationary signal that satisfies the Dirichlet criteria, where it is proven to be applicable. For example, the Fourier Series for signal x(t) = 3/2 that satisfies these conditions for period T=0 (number of discontinuities is 0, number of extrema points is 0 and the integral over period T is finite and equals 0), returns the constant component 3/2 only, which contains all knowledge about the signal oscillatory behavior, namely its absence. The exclusive heritage use of the Fourier series functions, the trigonometric functions of sine and cosine, also has the following three reasons . Given that we want a time invariant representation of signals, since there is usually no natural origin of time, leads to trigonometric functions that are the eigenfunctions of time translation. Linear systems also have the same eigenfunctions – the complex exponentials that are equivalent to the real trigonometric functions. The third good reason for the Fourier functions is that the synthesis of the band limited physical signal from equally spaced samples taken at a rate of at least twice higher than the signal’s highest frequency is simple to understand as a consequence of the Nyquist sampling theorem.
^ , and direct application of the Fourier spectrum analysis may lead to undesirable affects and unrelated physical interpretation. There is presently no engineering tool for systematic spectrum analysis and synthesis of nonlinear and nonstationary data. Nonlinear and nonstationary processes are complex processes that evolve in time, such as speech or music and their properties are statistically non-invariant over time. A theoretical attempt in this endeavor was proposed by Dennis Gabor in 1946 in his “Theory of Communication” as depicted in the book on Gabor analysis and Gabor frames . However, an effective computational method using Gabor frames is yet to be developed.
Dr. Norden E. Huang recently proposed a novel ^ (EMD) computational method for non-linear and non-stationary signal analysis. The EMD output basis functions, the Intrinsic Mode Functions (IMFs), are derived from the data and are susceptible to the Hilbert Transform for spectrum analysis, the Hilbert-Huang Transform (HHT) , , . The Implementation of the Digital Hilbert Transform is using the FFT. This paper describes the development of a novel engineering tool, the HHT Data Processing System that implements the HHT and allows a user to make use of HHT similar to the FFT for spectrum analysis of nonlinear and nonstationary data. The theoretical foundations of the HHT and the heritage FFT methods used in HHTDPS are discussed below.
|An Introduction to Hilbert-Huang Transform: a plea for Adaptive Data Analysis Norden E. Huang. Research Center for Adaptive Data Analysis. National Central University Часть Сущность преобразования. Обработка и анализ данных|
Норден E. Хуанг. Исследовательский центр адаптивного анализа данных. Национальный Центральный Университет
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