function?[u?u_hat?omega]?=?VMD（signal?alpha?tau?K?DC?init?tol）

%?Variational?Mode?Decomposition

%?Authors:?Konstantin?Dragomiretskiy?and?Dominique?Zosso

%?zosso@math.ucla.edu?---?http://www.math.ucla.edu/~zosso

%?Initial?release?2013-12-12?（c）?2013

%

%?Input?and?Parameters:

%?---------------------

%?signal??-?the?time?domain?signal?（1D）?to?be?decomposed

%?alpha???-?the?balancing?parameter?of?the?data-fidelity?constraint

%?tau?????-?time-step?of?the?dual?ascent?（?pick?0?for?noise-slack?）

%?K???????-?the?number?of?modes?to?be?recovered

%?DC??????-?true?if?the?first?mode?is?put?and?kept?at?DC?（0-freq）

%?init????-?0?=?all?omegas?start?at?0

%????????????????????1?=?all?omegas?start?uniformly?distributed

%????????????????????2?=?all?omegas?initialized?randomly

%?tol?????-?tolerance?of?convergence?criterion;?typically?around?1e-6

%

%?Output:

%?-------

%?u???????-?the?collection?of?decomposed?modes

%?u_hat???-?spectra?of?the?modes

%?omega???-?estimated?mode?center-frequencies

%

%?When?using?this?code?please?do?cite?our?paper:

%?-----------------------------------------------

%?K.?Dragomiretskiy?D.?Zosso?Variational?Mode?Decomposition?IEEE?Trans.

%?on?Signal?Processing?（in?press）

%?please?check?here?for?update?reference:?

%??????????http://dx.doi.org/10.1109/TSP.2013.2288675

?

?

?

%----------?Preparations

?

%?Period?and?sampling?frequency?of?input?signal

save_T?=?length（signal）;

fs?=?1/save_T;

?

%?extend?the?signal?by?mirroring

T?=?save_T;

f_mirror（1:T/2）?=?signal（T/2:-1:1）;

f_mirror（T/2+1:3*T/2）?=?signal;

f_mirror（3*T/2+1:2*T）?=?signal（T:-1:T/2+1）;

f?=?f_mirror;

?

%?Time?Domain?0?to?T?（of?mirrored?signal）

T?=?length（f）;

t?=?（1:T）/T;

?

%?Spectral?Domain?discretization

freqs?=?t-0.5-1/T;

?

%?Maximum?number?of?iterations?（if?not?converged?yet?then?it?won‘t?anyway）

N?=?500;

?

%?For?future?generalizations:?individual?alpha?for?each?mode

Alpha?=?alpha*ones（1K）;

?

%?Construct?and?center?f_hat

f_hat?=?fftshift（（fft（f）））;

f_hat_plus?=?f_hat;

f_hat_plus（1:T/2）?=?0;

?

%?matrix?keeping?track?of?every?iterant?//?could?be?discarded?for?mem

u_hat_plus?=?zeros（N?length（freqs）?K）;

?

%?Initialization?of?omega_k

omega_plus?=?zeros（N?K）;

switch?init

????case?1

????????for?i?=?1:K

????????????omega_plus（1i）?=?（0.5/K）*（i-1）;

????????end

????case?2

????????omega_plus（1:）?=?sort（exp（log（fs）?+?（log（0.5）-log（fs））*rand（1K）））;

????otherwise

???