If we want to have the frequency vs. time, or frequency content of a signal, we can use wavelet transform or Hilbert-Huang transform. What are the differences and advantages or disadvantages of these methods?
HH transform (also called empirical mode decomposition; EMD) and wavelet convolution are not interchangeable. They are quite different from each other, and whether to use wavelet or EMD depends completely on the question you are trying to answer. If you want an overview of the spectral changes in power over time, use wavelet transform and *not* EMD. A wavelet convolution is ideal for finding a nice balance between temporal and frequency precision. Wavelet convolution, filter-Hilbert (i.e., bandpass filtering and then applying the Hilbert transform), and short-time FFT are also conceptually and mathematically very similar to each other, and they are all different from EMD.
EMD, on the other hand, is an iterative procedure that extracts oscillatory-like features from the data. It provides only certain frequencies (specifically, those that have a lot of power and are ~1/2 the frequency of the previous frequency that was extracted) that cannot be specified in advance by you. The main advantage of EMD over wavelet convolution is the ability to estimate subtle changes in frequency (this is obtained via the first temporal derivative of the phase angle time series, scaled by the sampling rate and 2*f*pi). Estimating instantaneous frequency from a wavelet convolution is suboptimal because of frequency smoothing, and because wavelet convolution assumes frequency stationarity during the time span of the wavelet. EMD is also poorly suited for detecting relative suppressions of power at a specific frequency.
You wrote above "frequency vs. time." If you want a plot of instantaneous frequency by time, definitely go for EMD and not wavelet convolution. If you want a general characterization of the temporal changes in spectral dynamics of the signal, definitely go for wavelet convolution.
http://mitpress.mit.edu/9780262019873
Helow.
It is stated that they both split a signal into frequency bands
(Flandrin, P. Rilling, G. Goncalves, P. , Empirical mode decomposition as a filter bank Signal Processing Letters, IEEE, Volume: 11, Issue: 2, Part 1)
HHtransform is very very difficult to intrpret when analizing a wide band signal, for instance a 50khz meshing gearbox vibration response and particularly its instannteneous frequency. Its amplitude decomposition performance is better but cannot find differences comparing it to envelop analysis.
Michael Feldman, Analytical basics of the EMD: Two harmonics decomposition, Mechanical Systems and Signal Processing Volume 23 , Issue 7, October 2009, Pages 2059-2071, an excellent theoretical analysis on the limits of HHtransform.
Y. Kopsinis, S. McLaughlin, “Development of EMD-based Denoising
Methods Inspired by Wavelet Thresholding,” IEEE Trans. on Signal
Processing, VOL. 57, NO. 4, APRIL 2009. gives a practical application for HHtransform
In general in my view it provides great academic excersises to delve into digiital signal processing. Practical applications thogh (my view ) not so much. And (in my view) not very different from wavelet decomposition, both not very different from a practical view than a windowed short time fourier approach.
Wavelet transform convolves a signal with a predefined mother wavelet to decompose a signal.. The choice of the mother wavelet is usually dependent on the type of data you are dealing with.
HHT on the other hand does not require any convolution of the signal with a predefined basis function or mother wavelet. The process of decomposition is totally data-driven.
In my research, I use wavelet transform when I have a-priori information about the data I am dealing with. If the data is totally random and difficult to characterize, try using the HHT.
Thank you very much for the answers,
But, if this transforms aren't much different of each other, so why they have been invented? I mean, what HHT can do that Wavelet can not?
Of course Adu-Gyamfi is right. HHT was indeed designed to have more flexibility regarding radom signals and this is a potential of HHT.
In my view, what all signal transforms do is to adapt to geometric characteristics of a signal. All transforms are blind to the physical mechanism or the actual information which is hidden in the raw signal.
Most papers that i have read regarding transform comparisons are academic exercises. Some very good such as the work of Feldman. He knows his stuff.
Whether a transform performs better than another requires an expert to check it. Always test the result of a transform.
Just pick a transform that does the job sufficiently well for you. In my view there is no optimal answer.
Again all the latter are my view.
HH transform (also called empirical mode decomposition; EMD) and wavelet convolution are not interchangeable. They are quite different from each other, and whether to use wavelet or EMD depends completely on the question you are trying to answer. If you want an overview of the spectral changes in power over time, use wavelet transform and *not* EMD. A wavelet convolution is ideal for finding a nice balance between temporal and frequency precision. Wavelet convolution, filter-Hilbert (i.e., bandpass filtering and then applying the Hilbert transform), and short-time FFT are also conceptually and mathematically very similar to each other, and they are all different from EMD.
EMD, on the other hand, is an iterative procedure that extracts oscillatory-like features from the data. It provides only certain frequencies (specifically, those that have a lot of power and are ~1/2 the frequency of the previous frequency that was extracted) that cannot be specified in advance by you. The main advantage of EMD over wavelet convolution is the ability to estimate subtle changes in frequency (this is obtained via the first temporal derivative of the phase angle time series, scaled by the sampling rate and 2*f*pi). Estimating instantaneous frequency from a wavelet convolution is suboptimal because of frequency smoothing, and because wavelet convolution assumes frequency stationarity during the time span of the wavelet. EMD is also poorly suited for detecting relative suppressions of power at a specific frequency.
You wrote above "frequency vs. time." If you want a plot of instantaneous frequency by time, definitely go for EMD and not wavelet convolution. If you want a general characterization of the temporal changes in spectral dynamics of the signal, definitely go for wavelet convolution.
http://mitpress.mit.edu/9780262019873
Dear Dr. Cohen,
Thank you very much for your comprehensive answer.
It has been known for quite some time that the Hilbert transform of a wavelet is again
a wavelet.
i thnk using of each is depended to you problems condition. for example wavelet can be used for problem with determinestic condition and HHT can be used for un determinestic problems or random conditions
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