# Empirical Mode Decomposition and Sparsity

In what sense does empirical mode decomposition (EMD) bring out the sparsity in a signal?

For instance, if I had a signal $f$ and I broke it down into $n$ intrinsic mode functions (IMF), what should I do to get a representation that has very few nonzero components?

Generally when speaking of sparsity, we say that a signal is sparse in some basis. Since IMFs do not have a fixed basis, in what sense can they be (if at all) sparse?

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EMD? Are you referring to the Hilbert-Huang Transform? – Mohammad Oct 11 '12 at 20:14
Yes, I'm sorry, I should have mentioned it before. – Jaidev Deshpande Oct 11 '12 at 20:17

The idea of being sparse in a certain basis can be extended and one can talk about being sparse in a frame, or even, being sparse in a dictionary. Now we may lose orthogonality, have redundant signals in our dictionary and no longer have unique signal expansions; which is OK if we only care about sparsity.

The EMD expansion is sparse in the sense that you can express a signal as a linear combination of just a few IMFs.

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So if I throw away a few IMFs, (suppose the residue) then that would be called a sparse representation? Also, what do you mean by a dictionary? – Jaidev Deshpande Oct 12 '12 at 20:48
Yes. By dictionary I mean an arbitrary collection of functions (called atoms'') that you can use as building blocks by taking their linear combination for creating other functions. – Atul Ingle Oct 12 '12 at 21:06
Right. The decomposition of certain signal in 'arbitrary' bases would be sparse (eg, a stationary signal would be sparse in Fourier bases). Can the same be done with IMFs by discarding a few of them? Which are some good methods to identify the redundant IMFs? (PCA, for instance?) – Jaidev Deshpande Oct 12 '12 at 21:13
If there are correlations in the IMFs, their PCA should give you a set of orthogonal vectors which span the same space as the original IMFs. Also note that there is generally a difference between signals which are compressible (exhibiting an exponential decay of coefficient magnitude in a target basis) and those which are sparse (having only a few non-zero entries). Compressible signals can be approximated by sparse signals by retaining coefficients with the top $K$ magnitude. This would introduce a standard error term of $||f - f_K||$ into whatever you're doing. – Eric Oct 12 '12 at 21:26