Won't Hold Still
Frequency analysis with the Hilbert-Huang TransformBill Weaver, Ph.D
We start to look at vectors and their importance to motion analysis around chapter three of the freshman physics text. By that time, the related concepts of position, displacement, velocity and acceleration have been introduced, and we start with simple examples of constant velocity and constant acceleration, ultimately advancing to variable and non-linear velocity and acceleration problems. With the aid of graph slope and differential calculus, the class is able to grasp the concepts of instantaneous velocity and acceleration at a single point in time and space while the very definitions of velocity and acceleration require displacement in those dimensions. Vector notation allows us to decompose complex three-dimensional motion into orthogonal x, y and z components and motion analysis proceeds by analyzing each of the components individually. These high-level concepts have been around for such a long time that they are found in the introductory chapters of a first course in physics.
After proceeding through concepts of force and energy, we eventually examine variations in energy as a function of time, leading to the study of waves. While energy oscillations have amplitude, frequency and phase, we often are able to detect only their amplitude as a function of time using an appropriate transducer. We must employ data analysis to recover information about their phase and frequency. The Fourier Transform (FT) is a popular transform used to generate plots of energy as a function of frequency. The resulting spectra represent the individual sine waves that can be summed to produce the observed amplitude-time information. Unfortunately, while sine waves continue indefinitely, real waves most often do not. The FT also assumes the waves are not changing in frequency over time (i.e., are stationary), and it expects the detected amplitude-time information to be linear. When these conditions are not met, spurious high-frequency harmonics appear in the spectra.
The Wavelet Transform (WT) overcomes some of the limitations of the FT by replacing its continuous sine wave basis function by a discrete basis function or "wavelet," allowing for regional frequency analysis. The WT is applied over short time windows to produce an energy-frequency-time plot. With the correct selection of time window size, the WT can accommodate gradually varying non-stationary oscillations. Like the FT, the WT assumes the amplitude data is linear as it calculates the wavelets that are summed to generate the observed data within each time window. While adjusted for best fit, the size of the time window and the form of the basis wavelet must be chosen before the data is analyzed.
Faced with the task of analyzing non-stationary and non-linear oscillations produced by water waves, atmospheric winds and earthquakes, Norden E. Huang of the NASA Goddard Space Flight Center along with colleagues from NOAA, the Naval Research Laboratory, Naval Surface Warfare Center and academia began the development of a time-frequency transform capable of handling non-stationary and non-linear oscillations. Their radically new transform, the Hilbert-Huang Transform (HHT) is based on the early chapters of our physics texts. While the FT deals with integrating global average frequencies, and the WT deals with integrating regional average frequencies, the HHT is based on differentiated local instantaneous frequencies. Since a "basis function with instantaneous frequency" does not exist, the HHT is not an extension of either FT or WT analysis.
Much like instantaneous velocity can be calculated by drawing a tangent line on a position vs. time plot, the HHT derives its instantaneous frequency values by analyzing the entire amplitude vs. time plot graphically. A cubic spline is drawn through the local maxima, a second cubic spline is drawn through the local minima and the average of these two lines is subtracted from the original data. This "envelope and subtract" process, termed Empirical Mode Decomposition (EMD), is continued until the resulting waveform is symmetric with respect to the local zero mean and the number of zero crossings and extrema are the same or differ by one. When these conditions are satisfied, the waveform is said to be an Intrinsic Mode Function (IMF) representing a component mode of oscillation within the data.
The IMF can be a broadband signal, both amplitude and frequency modulated, non-stationary and non-linear. As the slope of a tangent line reveals the instantaneous velocity, the Hilbert Transform of the IMF yields the mode's instantaneous frequency as a function of time. Once an IMF is obtained, it is subtracted from the original data and EMD continues on the resultant until all of the component IMFs are discovered. The IMFs are obtained without a priori choice of basis functions and time window width, are not spurious results of the transform process, and represent actual physical oscillations.
The HHT was named the 2003 NASA Government Invention of the Year and is being applied in many areas of frequency analysis including NVH, medical, biometrics, finance, structural, communication, fluid dynamics and environmental applications. The increasing frequency of its use is anything but stationary.
Bill Weaver is an assistant professor in the Integrated Science, Business and Technology Program at LaSalle University. He may be contacted at firstname.lastname@example.org.