Nonstationary Signal Processing Hilbert-Huang Transform Joseph DePasquale 22 Mar 07

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

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

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

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

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

EMD stopping criterion

Hilbert Spectral Analysis (HSA) (1) (2)

HSA cont. (3) (4)

HT as a filter Hilbert transform of cosine is sine

Phase Shift Example

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

Observations Pseudo-filter, only changes phase No effect on amplitude of the signal Signal and it’s HT are orthogonal Signal and it’s HT have identical energy

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

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. 903–995, [2] N. E. Huang, “Introduction to the Hilbert-Huang Transform and It’s Related Mathematical Problems,” in The Hilbert-Huang Transform and Its Applications, 2005, pp