Diff for "HHT based Instrumental Glitch Trigger Generation"

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HHT based Instrumental Glitch Trigger Generation

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 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.

	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

Goal

Preliminary Analysis of HHT

Trigger Generation Method

Simulation Results

Comparison to Klein-Welle and Other Methods

HHT based Instrumental Glitch Trigger Generation (last edited 2014-01-09 12:58:08 by johnoh)

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+ * 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 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 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
-Line 5:
+Line 8:
- * Introduction: Hilbert-Huang Transform (HHT)
+{{https://upload.wikimedia.org/math/1/8/a/18aba9375519cd1b75a761363eea55b7.png}},
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+Line 10:
-{{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 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 {{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}}. 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/9/d/e/9de1f9c2b24c4c2b3bab29243d825bc3.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.
 Line 12:
-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.
+   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.

||<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 ||