Nonstationary Signal Processing Hilbert-Huang Transform

1
Nonstationary Signal ProcessingHilbert-Huang
Transform

• Joseph DePasquale
• 22 Mar 07

2
Overview

• Data Analysis
• Empirical Mode Decomposition
• Visual Example
• Hilbert Spectral Analysis
• Conclusions

3
Data Analysis

• Traditional methods
• Linear
• Stationary
• Newer methods
• e.g. wavelet analysis
• a priori basis used for data analysis

4
Adaptive Basis

• Necessary for representation of non-linear (NL)
  and nonstationary (NS) data
• Basis is data dependent
• a posteriori
• HHT meets some of the requirements for NL and NS
  analysis

5
Hilbert-Huang Transform (HHT)

• Two parts
• Empirical mode decomposition (EMD)
• Hilbert spectral analysis (HSA)
• Tested and validated exhaustively
• Empirical
• HHT provides sharper results than traditional
  methods of analysis
• Mathematical problems

6
Empirical Mode Decomposition

• Decompose a signal into intrinsic mode functions
  (IMF)
• IMF
• Defined by two criterion
• Signal represents simple oscillatory mode
• IMFs contain statistically significant
  information
• Extract this information through HSA

7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
EMD stopping criterion
13
Hilbert Spectral Analysis (HSA)
(1)
(2)
14
HSA cont.
(3)
(4)
15
HT as a filter
Hilbert transform of cosine is sine
16
Phase Shift Example

• www.adacs.com/menu/art/dsp_hilbert.gif

17
HT Properties

• The Hilbert transform of a constant is zero
• The Hilbert transform of a Hilbert transform is
  the negative of the original function
• A function and its Hilbert transform are
  orthogonal over the infinite interval
• The Hilbert transform of a real function is a
  real function
• The Hilbert transform of a sine function is a
  cosine function, the Hilbert transform of a
  cosine function is the negative of the sine
  function

18
Observations

• Pseudo-filter, only changes phase
• No effect on amplitude of the signal
• Signal and its HT are orthogonal
• Signal and its HT have identical energy

19
Conclusion

• Empirical tests indicate HHT is a superior tool
  for time-frequency analysis
• Employs an adaptive basis
• Mathematical theory not complete
• EMD is used to extract IMF
• HSA is used to find the instantaneous frequency
  of the individual IMF

20
References

• 1 N. E. Huang et. al, The empirical mode
  decomposition and the Hilbert spectrum for
  nonlinear and nonstationary time series
  analysis, Proc. Roy. Soc. Lond., vol. A 454, pp.
  903995, 1998.
• 2 N. E. Huang, Introduction to the
  Hilbert-Huang Transform and Its Related
  Mathematical Problems, in The Hilbert-Huang
  Transform and Its Applications, 2005, pp. 1-26.