www.pudn.com > hilbert.zip > emd_lily02.m, change:2008-05-14,size:6902b

```% EMD.M
%
% G. Rilling, July 2002
%
% computes EMD (Empirical Mode Decomposition) according to:
%
% N. E. Huang et al., "The empirical mode decomposition and the
% Hilbert spectrum for non-linear and non stationary time series analysis,"
% Proc. Royal Soc. London A, Vol. 454, pp. 903-995, 1998
%
% with variations reported in:
%
% G. Rilling, P. Flandrin and P. Gon鏰lv鑣
% "On Empirical Mode Decomposition and its algorithms"
% IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing
% NSIP-03, Grado (I), June 2003
%
% stopping criterion for sifting :
%   at each point : mean amplitude < threshold2*envelope amplitude
%   &
%   mean of boolean array ((mean amplitude)/(envelope amplitude) > threshold) < tolerance
%   &
%   |#zeros-#extrema|<=1
%
% inputs:  - x : analysed signal (line vector)
%          - t (optional) : sampling times (line vector) (default : 1:length(x))
%          - stop (optional) : threshold, threshold2 and tolerance (optional)
%                              for sifting stopping criterion
%                              default : [0.05,0.5,0.05]
%          - tst (optional) : if equals to 1 shows sifting steps with pause
%                             if equals to 2 no pause
%
% outputs: - imf : intrinsic mode functions (last line = residual)
%          - ort : index of orthogonality
%          - nbits : number of iterations for each mode
%
% calls:   - extr (finds extrema and zero-crossings)
%          - io : computes the index of orthogonality

function [imf,ort,nbits] = emd_lily0(x,t,stop,tst);
fs=0.001;
% default for stopping
defstop = [0.05,0.5,0.05];

if(nargin==1)
t = 1:length(x);
stop = defstop;
tst = 0;
end

if(nargin==2)
stop = defstop;
tst = 0;
end

if (nargin==3)
tst=0;
end

S = size(x);
if ((S(1) > 1) & (S(2) > 1)) | (length(S) > 2)
error('x must have only one row or one column')
end

if S(1) > 1
x = x';
end

S = size(t);
if ((S(1) > 1) & (S(2) > 1)) | (length(S) > 2)
error('t must have only one row or one column')
end

if S(1) > 1
t = t';
end

if (length(t)~=length(x))
error('x and t must have the same length')
end

S = size(stop);
if ((S(1) > 1) & (S(2) > 1)) | (S(1) > 3) | (S(2) > 3) | (length(S) > 2)
error('stop must have only one row or one column of max three elements')
end

if S(1) > 1
stop = stop';
S = size(stop);
end

if S(2) < 3
stop(3)=defstop(3);
end

if S(2) < 2
stop(2)=defstop(2);
end

sd = stop(1);
sd2 = stop(2);
tol = stop(3);

if tst
figure
end

% maximum number of iterations
MAXITERATIONS=2000;

% maximum number of symmetrized points for interpolations
NBSYM = 2;

lx = length(x);

sdt(lx) = 0;
sdt = sdt+sd;
sd2t(lx) = 0;
sd2t = sd2t+sd2;

% maximum number of extrema and zero-crossings in residual
ner = lx;
nzr = lx;

r = x;
imf = [];
k = 1;

% iterations counter for extraction of 1 mode
nbit=0;

% total iterations counter
NbIt=0;

while ner > 2

% current mode
m = r;

% mode at previous iteration
mp = m;

sx = sd+1;

% tests if enough extrema to proceed
test = 0;

[indmin,indmax,indzer] = extr(m);
lm=length(indmin);
lM=length(indmax);
nem=lm + lM;
nzm=length(indzer);

j=1;

% sifting loop
while ( mean(sx > sd) > tol | any(sx > sd2) | (abs(nzm-nem)>1)) & (test == 0) & nbit<MAXITERATIONS

if(nbit>MAXITERATIONS/5 & mod(nbit,floor(MAXITERATIONS/10))==0)
disp(['mode ',int2str(k),' nombre d iterations : ',int2str(nbit)])
disp(['stop parameter mean value : ',num2str(s)])
end

% boundary conditions for interpolations :包络极值延拓法
if (lm==1)&(lM==1)
k1=2*abs(indmax(1)-indmin(1));
end
if indmax(1) < indmin(1)
k1=indmax(2)-indmax(1);
else if indmax(1) > indmin(1)
k1=indmin(2)-indmin(1);
end
end
tlmax2=t(indmax(1))-2*k1*(1/fs);
mlmax2=m(indmax(1));
tlmax1=t(indmax(1))-k1*(1/fs);
mlmax1=m(indmax(1));
tlmin2=t(indmin(1))-2*k1*(1/fs);
mlmin2=m(indmin(1));
tlmin1=t(indmin(1))-k1*(1/fs);
mlmin1=m(indmin(1));

if (lm==1)&(lM==1)
k2=2*abs(indmax(1)-indmin(1));
end
if indmax(lM) > indmin(lm)
k2=indmax(lM)-indmax(lM-1);
else if indmax(lM) < indmin(lm)
k2=indmin(lm)-indmin(lm-1);
end
end
trmax2=t(indmax(lM))+2*k2*(1/fs);
mrmax2=m(indmax(lM));
trmax1=t(indmax(lM))+k2*(1/fs);
mrmax1=m(indmax(lM));
trmin2=t(indmin(lm))+2*k2*(1/fs);
mrmin2=m(indmin(lm));
trmin1=t(indmin(lm))+k2*(1/fs);
mrmin1=m(indmin(lm));

% definition of envelopes from interpolation

envmax = interp1([tlmax2 tlmax1 t(indmax) trmax1 trmax2],[mlmax2 mlmax1 m(indmax) mrmax1 mrmax2],t,'spline');
envmin = interp1([tlmin2 tlmin1 t(indmin) trmin1 trmin2],[mlmin2 mlmin1 m(indmin) mrmin1 mrmin2],t,'spline');

envmoy = (envmax + envmin)/2;

m = m - envmoy; %极值对称延拓法

[indmin,indmax,indzer] = extr(m);
lm=length(indmin);
lM=length(indmax);
nem = lm + lM;
nzm = length(indzer);

% evaluation of mean zero
sx=2*(abs(envmoy))./(abs(envmax-envmin));
s = mean(sx);

% display

if tst
subplot(4,1,1)
plot(t,mp);hold on;
plot(t,envmax,'--k');plot(t,envmin,'--k');plot(t,envmoy,'r');

title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' before sifting']);
set(gca,'XTick',[])
hold  off

subplot(4,1,2)
plot(t,sx)
hold on
plot(t,sdt,'--r')
plot(t,sd2t,':k')
title('stop parameter')
set(gca,'XTick',[])
hold off

subplot(4,1,3)
plot(t,m)
title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' after sifting']);
set(gca,'XTick',[])

subplot(4,1,4);
plot(t,r-m)
title('residue');
disp(['stop parameter mean value : ',num2str(s)])
if tst == 2
pause(0.01)
else
pause
end

end

% end loop : stops if not enough extrema
if nem < 3
test = 1;
end

mp = m;
nbit=nbit+1;
NbIt=NbIt+1;

if(nbit==(MAXITERATIONS-1))
warning(['forced stop of sifting : too many iterations... mode ',int2str(k),'. stop parameter mean value : ',num2str(s)])
end

end
imf(k,:) = m;
if tst
disp(['mode ',int2str(k),' enregistre'])
end
nbits(k) = nbit;
k = k+1;
r = r - m;
[indmin,indmax,indzer] = extr(r);
ner = length(indmin) + length(indmax);
nzr = length(indzer);
nbit=1;

if (max(r) - min(r)) < (1e-10)*(max(x) - min(x))
if ner > 2
warning('forced stop of EMD : too small amplitude')
else

disp('forced stop of EMD : too small amplitude')
end
break
end

end

imf(k,:) = r;
n=size(imf);
n=n(1);
ort = io(x,imf);

if tst
close
end

```