MATLAB: How can fit the data by multiple damping cosine functions

cosine fitting damping

Hi there,
could you please help me out to fit my data via several damping consine functions like:
s(t)=A(cos(B*t)+cos(C*t))*exp(-D*t)+m
Thanks a lot in advance.
All the bests,
PJ

Best Answer

This is as close as I can get to a decent fit: 
D = load('data_fft_peak6.mat');
t = D.data_fft_peak(:,1);
s = D.data_fft_peak(:,2);
figure
plot(t, s)
grid
L = numel(t);
Ts = mean(diff(t));
Fs = 1/Ts;
Fn = Fs/2;
N = 2^(nextpow2(L)+2);
FTs = fft(s,N)/L;
Fv = linspace(0, 1, fix(numel(FTs)/2)+1)*Fn;
Iv = 1:numel(Fv);
% [pks,locs] = findpeaks(abs(FTs(Iv))*2, 'MinPeakProminence',0.001);
[pks,locs] = findpeaks(abs(FTs(Iv))*2, 'MinPeakHeight',0.02);
figure
plot(Fv, abs(FTs(Iv))*2)
hold on
plot(Fv(locs), abs(FTs(locs))*2, 'r^')
hold off
grid
pklbls = compose('\\leftarrow Freq = %8.5f Hz\n     Ampl = %8.5f', [Fv(locs).', pks]);
text(Fv(locs), pks, pklbls, 'HorizontalAlignment','left', 'VerticalAlignment','top')
% objfcn = @(b,t) b(1).*cos(b(2)*t) + b(3).*cos(b(4).*t) + b(5).*cos(b(6).*t) .* b(7).*t.*exp(b(8).*t);
% B0 = [pks(1) Fv(locs(1))*2*pi pks(2) Fv(locs(2))*2*pi pks(3) Fv(locs(3))*2*pi 0.5 -0.01];
objfcn = @(b,t) b(1).*cos(b(2)*t) + b(3).*cos(b(4).*t) .* b(5).*t.*exp(b(6).*t);
B0 = [pks(1) Fv(locs(1))*2*pi pks(2) Fv(locs(2))*2*pi 0.5 -0.01].';
B = lsqcurvefit(objfcn, B0, t, s) 
fitfcn = objfcn(B, t);
figure
plot(t, s)
hold on
plot(t, fitfcn, '-r')
hold off
grid
Note — I included an extended version of ‘objfcn’ for you to experiment with.
The estimated parameters are: 
B = 
   0.022930977208190
   0.106707811047044
  -0.000717587879530
   0.112740321550530
   1.119470956816299
  -0.002453356895356
EDIT — (29 Dec 2020 at 17:29)
Also experiment with: 
[pks,locs] = findpeaks(abs(FTs(Iv))*2, 'MinPeakProminence',0.008);
and: 
objfcn = @(b,t) b(1).*cos(b(2)*t) + b(3).*cos(b(4).*t) .* b(7).*t.*exp(b(8).*t) + b(5).*cos(b(6).*t);
B0 = [pks(1) Fv(locs(1))*2*pi pks(2) Fv(locs(2))*2*pi pks(3) Fv(locs(3))*2*pi 0.5 -0.01].';
producing: 
B = 
   0.007599194766653
   0.010943887298747
  -0.000688462858777
   0.112676604541936
   0.035329767711197
   0.118358798142968
   1.119527877005960
  -0.002418300550008
.
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