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| ==== Project Description ==== * Introduction: Hilbert-Huang Transform (HHT) HHT is a recently suggested empirical data transform based on adaptive bases. So it is very useful for analyzing non-linear and/or non-stationary data. The HHT consists of two main parts: 1) empirical mode decomposition (EMD) 2) Hilbert transform. The EMD can decompose the original data into some appropriate data sets that contributes to some frequency bands. Basically the EMD is described by a sifting function defined as {{CodeCogsEqn.png}} ==== Goal ==== |
==== Project Description and Goal ==== ==== Introduction: Hilbert-Huang Transform (HHT) ==== {{attachment:emd2.png||align="left"}} HHT is a recently suggested empirical data transform based on adaptive bases. So it is very useful for analyzing non-linear and/or non-stationary data. The HHT consists of two main parts: 1) empirical mode decomposition (EMD) 2) Hilbert spectral analysis (HSA). The EMD can decompose the original data into some appropriate data sets that contributes to some frequency bands. Basically the EMD is described by a sifting processes with {{https://upload.wikimedia.org/math/e/e/4/ee4471f673afaddfc791f692cf250f6b.png}}, which shows that the EMD performs the subtraction between the original data (or the previously generated data) and its average of envelopes, repeatedly, called '''''intrinsic mode functions (IMF)''''', expressed by {{https://upload.wikimedia.org/math/7/e/c/7ec2bae54bcdd79fab725ef8d6c5314c.png}} and {{https://upload.wikimedia.org/math/6/5/3/65349fc2c693c1b809562b59f4b5e172.png}}. And the normalized squared difference (NSD) between two successive sifting operations is defined as {{https://upload.wikimedia.org/math/1/8/a/18aba9375519cd1b75a761363eea55b7.png}}, which should be small. The stoppage criterion of this process is determined by comparing between the NSD and the predetermined value - if NSD is smaller than a predetermined value, the process is stopped. The Hilbert transform of the ''EMDed data'' consists of the amplitude and phase parts and we see that the original data can be expressed by the summation of the whole IMFs, which is {{https://upload.wikimedia.org/math/6/f/d/6fdad0aefbac705612d1e8541fcd9eef.png}}. Comparing to the (Fast) Fourier Transform (FFT), {{https://upload.wikimedia.org/math/2/7/7/277cc4bee5968c6ca2d4c78747d18689.png}}, we easily see that the HHT deals with the time-variant amplitude and phase data with adaptive bases. We compare the HHT to other two different transforms in the following table. ||<bgcolor="#E0E0FF"> ||<bgcolor="#E0E0FF"> Fourier ||<bgcolor="#E0E0FF"> Wavelet ||<bgcolor="#E0E0FF"> Hilbert-Huang || ||<bgcolor="#E0E0FF">Basis || a priori || a priori || Adaptive || ||<bgcolor="#E0E0FF">Frequency || Integral transform: Global || Integral transform: Regional || Differentiation: Local|| ||<bgcolor="#E0E0FF">Presentation || Energy-Frequency || Energy-Time-Frequency || Energy-Time-Frequency|| ||<bgcolor="#E0E0FF"> Nonlinearity || No || No || Yes || ||<bgcolor="#E0E0FF"> Non-stationarity || No || Yes || Yes || ||<bgcolor="#E0E0FF"> Uncertainty || Yes || Yes || No || ||<bgcolor="#E0E0FF">Harmonics || Yes || Yes || No || |
HHT based Instrumental Glitch Trigger Generation
Project Description and Goal
Introduction: Hilbert-Huang Transform (HHT)
HHT is a recently suggested empirical data transform based on adaptive bases. So it is very useful for analyzing non-linear and/or non-stationary data. The HHT consists of two main parts: 1) empirical mode decomposition (EMD) 2) Hilbert spectral analysis (HSA). The EMD can decompose the original data into some appropriate data sets that contributes to some frequency bands. Basically the EMD is described by a sifting processes with
, which shows that the EMD performs the subtraction between the original data (or the previously generated data) and its average of envelopes, repeatedly, called intrinsic mode functions (IMF), expressed by 
and 
. And the normalized squared difference (NSD) between two successive sifting operations is defined as
,
which should be small. The stoppage criterion of this process is determined by comparing between the NSD and the predetermined value - if NSD is smaller than a predetermined value, the process is stopped.
The Hilbert transform of the EMDed data consists of the amplitude and phase parts and we see that the original data can be expressed by the summation of the whole IMFs, which is
. Comparing to the (Fast) Fourier Transform (FFT),
, we easily see that the HHT deals with the time-variant amplitude and phase data with adaptive bases. We compare the HHT to other two different transforms in the following table.
|
Fourier |
Wavelet |
Hilbert-Huang |
Basis |
a priori |
a priori |
Adaptive |
Frequency |
Integral transform: Global |
Integral transform: Regional |
Differentiation: Local |
Presentation |
Energy-Frequency |
Energy-Time-Frequency |
Energy-Time-Frequency |
Nonlinearity |
No |
No |
Yes |
Non-stationarity |
No |
Yes |
Yes |
Uncertainty |
Yes |
Yes |
No |
Harmonics |
Yes |
Yes |
No |
Preliminary Analysis of HHT
Trigger Generation Method
Simulation Results