Empirical Mode Decomposition-EMD

EMD was proposed by Norden E. Huang, and used for non-stationary and non-linear signal analysis.

Intuitively if we connect all the maxima of one signal, we will get its envelope, the same is true for minima. then signal can be decomposed into the mean of the two envelopes and the residue. If we do this operation continually, then we will finally get a signal without riding waves(baseline is zero, i.e. mean of envelopes will be zero).

1. EMD is a data driven algorithm, it decomposes the signal into IMF, which is not a pre-defined signal. IMF is generated by data itself.

2. The idea is : the signal is decomposed into local oscillations and its residues(can be viewed as a baseline) in some scale(it is not fixed). The osillations is high frequency part resides on the low frequency residues. If we do the same decomposition on the residues, then we can get another osillations with wider scale.

3. The algorithm is based on finding oscillations using local extrema.

1) identify all extrema of x(t)

2) interpolate between minima(resp. maxima), ending up with some envelope e_min(resp. e_max)

3) compute the mean m(t) = (e_min+e_max)/2

4) extract the detail d(t) = x(t)-m(t)

5)Iterate steps 1)-4) on the residual until the detail signal d(t) can be considered an IMF: c(t)=d(t).

6) iterate 1)-5) on the residual m(t) to extract all the IMFs. The stop criteria will be: the signal is constant or monotonic, or with only one extrema.

Definition of IMF: Functions with equal number of extrema and zero-crossings, and must be zero-mean defined by its maximas and minimas. This is for consideration of removing riding wave or baseline, and it has a good Hilbert Transform results, that is it is an good AM-FM signal in the form of a(t)exp( i*theta(t) ), and for IMF, its envelope defined by HT will make sense.

Intermittent Test: To select time scale(shorter than this will not be included) to avoid oscillations with different time scale(frequency) mixed with each other into same IMF.

How to use EMD:

1. EMD of signal

2. Spectral Analysis of IMFs

3. Selection of IMFs based on Spectral characteristic.

4. Reconstruction or partial reconstruction of signal for analysis.