% Time varying filtering based EMD
function imf = tvf_emd(x)
%input
% x: input signal x(t)
%output
% imf: resulting intrinsic mode functions
%%%%%%%%%%% preprocessing
THRESH_BWR=0.1; % inst bandwidth threshold (stopping criterion)
MAX_IMF=50; % max output imfs
BSP_ORDER=26; % b-spline order
end_flag=0;
imf=zeros(MAX_IMF,numel(x));
if size(x,1) > 1 % convert to row vector
x = x.';
end
temp_x=x;
t=1:numel(x);
for nimf=1:MAX_IMF
[indmin_x, indmax_x] = extr(temp_x); % is there enough extrema to continue
if nimf==MAX_IMF
imf(nimf,:)=temp_x;
nimf=nimf+1;
break;
end
if(numel([indmin_x, indmax_x])<4)
imf(nimf,:)=temp_x;
if numel(find(temp_x~=0))>0
nimf=nimf+1;
end
end_flag=1;
end
if end_flag==1
break;
end
num_padding = round(length(temp_x)*0.5);% padding number
y = temp_x;
flag_stopiter=0;
for iter=1:100
y = [fliplr(y(2:2+num_padding-1)) y fliplr(y(end-num_padding:end-1))]; % padding to deal with boundary effect (symmetric)
%y=extendsignal(y,num_padding); % padding to deal with boundary effect (match slope)
tt=1:numel(y);
ind_remov_pad=num_padding+1:numel(y)-num_padding;
%%%%%%%%%%%%% extract inst amp and inst freq using hilbert transform
[indmin_y, indmax_y] = extr(y);
indexC_y=sort([indmin_y, indmax_y]);
[instAmp0,instFreq0] = INST_FREQ_local(y);
%%%%%%%%%%%%%% divide y(t) into two parts and compute bis freq (cutoff freq)
[a1 f1 a2 f2 bis_freq instBWR avgFreq]=devide_y(y,instAmp0,instFreq0);
instBWR2=instBWR;
for j=1:2:numel(indexC_y)-2
ind=indexC_y(j):indexC_y(j+2);
instBWR2(ind)=mean(instBWR(ind));
end
bis_freq(instBWR2<THRESH_BWR)=1e-12;
bis_freq(bis_freq>0.5)=0.45; bis_freq(bis_freq<=0)=1e-12;
%%%%%%%%%%%% deal with mode mixing
bis_freq = anti_modemixing(y,bis_freq,ind_remov_pad,num_padding);
bis_freq=bis_freq(ind_remov_pad);
bis_freq = [fliplr(bis_freq(2:2+num_padding-1)) bis_freq fliplr(bis_freq(end-num_padding:end-1))];
bis_freq = anti_modemixing(y,bis_freq,ind_remov_pad,num_padding);
bis_freq=bis_freq(ind_remov_pad);
bis_freq = [fliplr(bis_freq(2:2+num_padding-1)) bis_freq fliplr(bis_freq(end-num_padding:end-1))];
%%%%%%%%%%%%%% stopping criteria
temp_instBWR=instBWR2(ind_remov_pad);
ind_start=round(numel(temp_instBWR)*0.05);% incase boundary effect
ind_end=round(numel(temp_instBWR)*0.95);
if (iter>=2 && mean(temp_instBWR(ind_start:ind_end))<THRESH_BWR+THRESH_BWR/4*iter) || iter>=6 || (nimf>1 && mean(temp_instBWR(ind_start:ind_end))<THRESH_BWR+THRESH_BWR/4*iter)
flag_stopiter=1;
end
if numel(find(temp_instBWR(ind_start:ind_end)>THRESH_BWR))/numel(instBWR2(ind_remov_pad))<0.2 % most of the signal are local narrowband
flag_stopiter=1;
end
%%%%%%%%%%%%%% obtain local mean using time varying filtering
phi=zeros(1,numel(bis_freq)); % build knots via h(t)
for i=1:numel(bis_freq)-1
phi(i+1)=phi(i)+2*pi*bis_freq(i);
end
[indmin_knot, indmax_knot] = extr(cos(phi));
indexC_knot=sort([indmin_knot, indmax_knot]);
if numel(indexC_knot)>2
pp_spline = splinefit(1:length(y),y,indexC_knot,BSP_ORDER,'p'); % TVF filtering with the extrema of h(t) as knots
localmean = ppval(pp_spline,1:length(y));
else
flag_stopiter=1; %not enough knots to perform filtering
end
if (max(abs(y(ind_remov_pad)-localmean(ind_remov_pad)))/min(abs(localmean(ind_remov_pad)))<1e-3) % prevent signal of very small amplitude to be extracted
flag_stopiter=1;
end
temp_residual=y-localmean;
temp_residual=temp_residual(ind_remov_pad);
temp_residual=temp_residual(round(numel(temp_residual)*0.1):end-round(numel(temp_residual)*0.1));
localmean2=localmean(ind_remov_pad);
localmean2=localmean2(round(numel(localmean2)*0.1):end-round(numel(localmean2)*0.1));
if abs(max(localmean2))/abs(max(instAmp0(ind_remov_pad)))<3.5e-2 || abs(max(temp_residual))/abs(max(instAmp0(ind_remov_pad)))<1e-2
flag_stopiter=1;
end
%%%%%%%% check stopping criteria
if flag_stopiter
imf(nimf,:)=y(ind_remov_pad);
temp_x=temp_x-y(ind_remov_pad);
break;
end
y=y-localmean;
y=y(ind_remov_pad);
end
end
imf(nimf:MAX_IMF,:)=[];
end
function output_cutoff = anti_modemixing(y,bis_freq,ind_remov_pad,num_padding)
org_bis_freq=bis_freq;
output_cutoff=bis_freq;
flag_intermitt=0;
t=1:numel(bis_freq);
intermitt=[];
%%%%% locate the maxima of the input signal
[indmin_y, indmax_y] = extr(y);
indexC_y=sort([indmin_y, indmax_y]);
zero_span=[];
%%%% preprocessing
for i=2:numel(indmax_y)-1
time_span=indmax_y(i-1):indmax_y(i+1);
if (max(bis_freq(time_span))-min(bis_freq(time_span)))/min(bis_freq(time_span))>0.25
zero_span=[zero_span time_span];
end
end
bis_freq(zero_span)=0;
%%%% find out all intermittences
diff_bis_freq=zeros(size(bis_freq));
for i=1:numel(indmax_y)-1
time_span=indmax_y(i):indmax_y(i+1);
if (max(bis_freq(time_span))-min(bis_freq(time_span)))/min(bis_freq(time_span))>0.25
intermitt=[intermitt indmax_y(i)];
diff_bis_freq(indmax_y(i))=bis_freq(indmax_y(i+1))-bis_freq(indmax_y(i));
end
end
ind_remov_pad([1:round(0.1*numel(ind_remov_pad)),round(0.9*numel(ind_remov_pad)):end])=[];
inters=intersect(ind_remov_pad,intermitt);
if numel(inters) >0
flag_intermitt=1;
end
%%%% find out floors
% plot(t(intermitt),bis_freq(intermitt),'r.',t,bis_freq,'b')
for i=2:numel(intermitt)-1
u1=intermitt(i-1);
u2=intermitt(i);
u3=intermitt(i+1); % check the derivative of cutoff frequency
if diff_bis_freq(u2)>0 % rising edge
bis_freq(u1:u2)=0;
end
if diff_bis_freq(u2)<0 % falling edge
bis_freq(u2:u3)=0;
end
end
temp_bis_freq=bis_freq;
temp_bis_freq(temp_bis_freq<1e-9)=0; %cutoff freq of very small value is considered to be zero (floor value)
temp_bis_freq=temp_bis_freq(ind_remov_pad);
temp_bis_freq = [fliplr(temp_bis_freq(2:2+num_padding-1)) temp_bis_freq fliplr(temp_bis_freq(end-num_padding:end-1))];
flip_bis_freq=fliplr(bis_freq);
if numel(find(temp_bis_freq>1e-9))>0 && numel(find(flip_bis_freq>1e-9,1,'first'))>0
temp_bis_freq(1)=bis_freq(find(bis_freq>1e-9,1,'first'));temp_bis_freq(end)=flip_bis_freq(find(flip_bis_freq>1e-9,1,'first')); %padding boundary for interpolation
else
temp_bis_freq(1)=bis_freq(1); temp_bis_freq(end)=bis_freq(end);
end
bis_freq=temp_bis_freq;
%%%% interpolate between peaks
if numel(t(bis_freq~=0))<2
return;
end
bis_freq = interp1(t(bis_freq~=0),bis_freq(bis_freq~=0) ,t,'pchip');
flip_bis_freq=fliplr(org_bis_freq);
if numel(find(org_bis_freq>1e-9))>0 && numel(find(flip_bis_freq>1e-9,1,'first'))>0
org_bis_freq(1)=org_bis_freq(find(org_b
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