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Empirical mode decomposition

`[imf,residual] = emd(X)`

`[imf,residual,info] = emd(X)`

`[___] = emd(___,Name,Value)`

`emd(___)`

`[___] = emd(___,`

estimates emd with additional options specified by one or more
`Name,Value`

)`Name,Value`

pair arguments.

`emd(___)`

plots the original signal, IMFs, and
residual signal as subplots in the same figure.

**Empirical Mode Decomposition**

`emd`

decomposes a signal *X(t)* into *k* number of intrinsic mode functions (IMF), and residual *r _{k}(t)* using the sifting process. A brief overview of the sifting process,
listed in [1] and [2], is as follows:

Find local maxima and minima for signal

*X(t)*to construct an upper envelope*s*, and a lower envelope_{+}(t)*s*._{-}(t)Compute mean envelope for

*i*iteration,^{th}*m*,_{k,i}(t)$${m}_{k,i}\left(t\right)=\frac{1}{2}\left[{s}_{+}\left(t\right)+{s}_{-}\left(t\right)\right]$$

With

*c*=_{k}(t)*X(t)*for the first iteration, subtract mean envelope from residual signal,$${c}_{k}\left(t\right)={c}_{k}\left(t\right)-{m}_{k,i}\left(t\right)$$

If

*c*does not match the criteria of an IMF, steps 4 and 5 are skipped. The procedure is iterated again at step 1 with the new value of_{k}(t)*c*._{k}(t)If

*c*matches the criteria of an IMF, a new residual is computed. To update the residual signal, subtract the_{k}(t)*k*IMF from the previous residual signal,^{th}$${r}_{k}\left(t\right)={r}_{k-1}\left(t\right)-{c}_{k}\left(t\right)$$

Then begin from step 1, using the residual obtained as a new signal

*r*, and store_{k}(t)*c*as an intrinsic mode function._{k}(t)

For *N* intrinsic mode functions, the original signal is represented
as,

$$X\left(t\right)={\displaystyle \sum _{i=1}^{N}{c}_{i}\left(t\right)+{r}_{N}\left(t\right)}$$

For more information about the sifting process, see [1] and [2].

`SiftRelativeTolerance`

is a Cauchy type stop criterion proposed in
[4]. Sifting stops when current
relative tolerance is less than `SiftRelativeTolerance`

. The current
relative tolerance is defined as,

$$R\text{e}lativeTolerance\triangleq \frac{{\Vert c{\left(t\right)}_{previous}-c{\left(t\right)}_{current}\Vert}^{2}}{{\Vert c{\left(t\right)}_{current}\Vert}^{2}}$$

Energy ratio is the ratio of the energy of the signal at the beginning of sifting and
the average envelope energy.[3] Decomposition stops when
current energy ratio is larger than `MaxEnergyRatio`

. For *k* IMFs, `EnergyRatio`

is defined as,

$$EnergyRatio\triangleq 10{\mathrm{log}}_{10}\left(\frac{{\Vert X\left(t\right)\Vert}^{2}}{{\Vert {r}_{k}\left(t\right)\Vert}^{2}}\right)$$

[1] Norden E. Huang, Zheng Shen,
Steven R. Long, Manli C. Wu, Hsing H. Shih, Quanan Zheng, Nai-Chyuan Yen, Chi Chao Tung,
Henry H. LiuProc. R. Soc. Lond. A. "The empirical mode decomposition and the Hilbert
spectrum for nonlinear and non-stationary time series analysis." *Proceedings
of the Royal Society of London. Series A: Mathematical, Physical and Engineering
Sciences* 1998. 454. 903-995. 10.1098/rspa.1998.0193.

[2] Rilling, G & Flandrin,
Patrick & Gonçalves, Paulo. "On empirical mode decomposition and its algorithms."
*IEEE-EURASIP workshop on nonlinear signal and image processing*
2003. NSIP-03. Grado, Italy. 8-11.

[3] Rato, R.T. & Ortigueira,
Manuel & Batista, Arnaldo. "On the HHT, its problems, and some solutions."
*Mechanical Systems and Signal Processing* 2008. 22. 1374-1394.
10.1016/j.ymssp.2007.11.028.

[4] Wang, Gang & Chen, Xianyao
& Qiao, Fang-Li & Wu, Zhaohua & Huang, Norden. "On Intrinsic Mode Function."
*Advances in Adaptive Data Analysis* 2010. 2. 277-293.
10.1142/S1793536910000549.