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The Empirical Mode Decomposition Method Sifting. Goal of Data Analysis. To define time scale or frequency. To define energy density. To define joint frequency-energy distribution as a function of time.
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Goal of Data Analysis • To define time scale or frequency. • To define energy density. • To define joint frequency-energy distribution as a function of time. • To do this, we need a AM-FM decomposition of the signal: X(t) = A(t) cosθ(t), where A(t) defines local energy and θ(t) defines the local frequency. This is a Generalized Fourier Expansion.
Need for Decomposition • Hilbert Transform (and all other IF computation methods) offers meaningful Instantaneous Frequency for IMFs. • For complicate data, there should be more than one independent component at any given time. • The decomposition should be adaptive in order to study data from nonstationary and nonlinear processes. • Frequency space operations are difficult to track temporal changes.
Why Hilbert Transform is not enough? Even though mathematicians told us that the Hilbert transform exists for all functions of Lp-class.
Problems on ‘Envelope’ A seemingly simple proposition but it is not so easy.
Observations • None of the two “envelopes” seem to make sense in term of Generalized Fourier Expansion (GFE): • The Hilbert transformed amplitude oscillates too much. • The line connecting the local maximum is almost the tracing of the data. • It turns out that, though Hilbert transform exists, the simple Hilbert transform does not make sense physically. • For “envelopes” to make sense in terms of GFE, the necessary condition for Hilbert transformed amplitude to make sense is for IMF.
Observations • For each IMF, the envelope in GFE will make sense. • For complicate data, we have to decompose it before attempting envelope construction. • To be able to determine the envelope is equivalent to AM & FM decomposition.
Observations • Even for this well behaved function, the amplitude from Hilbert transform does not serve as an “envelope” well. One of the reasons is that the function has two spectrum lines. • Hilbert Transform represents higher frequency better. • Complications for more complex functions are many. • Here, the empirical envelope seems reasonable.
Empirical Mode Decomposition • Mathematically, there are infinite number of ways to decompose a functions into a complete set of components. • The ones that give us more physical insight are more significant. • In general, the fewer the number of representing components, the higher the information content: Sparse representation. • The adaptive method will represent the characteristics of the signal better. • EMD is an adaptive method that can generate infinite many sets of IMF components to represent the original data.
Empirical Mode DecompositionSifting : to get one IMF component
Empirical Mode DecompositionSifting : to get one IMF component
Empirical Mode DecompositionSifting : to get all the IMF components
Empirical Mode DecompositionSifting : to get all the IMF components
Empirical Mode DecompositionSifting : to get all the IMF components
Empirical Mode DecompositionSifting : to get all the IMF components
Observations • All IMF components are the sums of spline functions. • We selected cubic natural spline tomaintain the maximum smoothness. • We will discuss spline function next time.
The Effects of Sifting • The first effect of sifting is to eliminate the riding waves : to make the number of extrema equals to that of zero-crossing. • The second effect of sifting is to make the envelopes symmetric. The consequence is to make the amplitudes of the oscillations more even.
Critical Parameters for EMD • The maximum number of sifting allowed to extract an IMF, N. • The criterion for accepting a sifting component as an IMF, the Stoppage criterion S. • Therefore, the nomenclature for the IMF are CE(N, S) : for extrema sifting CC(N, S) : for curvature sifting
The Stoppage Criteria : S and SD A. The S number : S is defined as the consecutive number of siftings, in which the numbers of zero-crossing and extrema are the same for these S siftings. B. If the mean is smaller than a pre-assigned value. C. Fixed sifting (iterating) time. D. SD is small than a pre-set value, where
Curvature Sifting Hidden Scales