# How to find the best parameters to fit damped oscillations curves

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Hello,

My goal is to fit the best these filtered data (attached) with the equation :

x(1)*exp(-x(2)*t).*sin(2*pi*x(3)*t+x(4)).

Here is my code (below) but this algorithm doesn't give a good fitting.

I don't know if it is a problem of parameters (for example, how to choose x0, ub and lb. I read some information on Matlab website but it doesn't help me)...

Thanks in advance for your help !

fun = @(x,t) x(1)*exp(-x(2)*t).*sin(2*pi*x(3)*t+x(4));

t=[0:1/1000:(1/1000*(length(data_TRT)-1))];

x0 = [2,1,1,1]

options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt');

lb = [0,0,0,-1]

ub = [2,100,100,1]

[x,resnorm] = lsqcurvefit(fun,x0,t,data_TRT,lb,ub,options)

figure

plot(t,data_TRT,'k-',t,fun(x,t),'r-')

legend('Experimental Data','Modeled Data')

title('Data and Fitted Curve')

xlabel('Temps (secondes)')

Damping_Nigg =x(2)

Frequence_Nigg =x(3)

##### 0 Comments

### Accepted Answer

Star Strider
on 22 Jan 2020

Edited: Star Strider
on 22 Jan 2020

Specifically:

D = load('Louise data_TRT_HELP.mat');

data = D.data_TRT;

t= linspace(0, numel(data), numel(data));

y=data;

Fs=500;

Fn = Fs/2; % Nyquist Frequency

L = size(t,2);

y = detrend(y); % Remove Linear Trend

fty = fft(y)/L;

Fv = linspace(0, 1, fix(L/2)+1)*Fn; % Frequency Vector

Iv = 1:length(Fv); % Index Vector

figure(1)

plot(Fv, abs(fty(Iv)))

grid

Wp = 30/Fn; % Passband (Normalised)

Ws = 40/Fn; % Stopband (Normalised)

Rp = 1; % Passband Ripple

Rs = 50; % Stopband Ripple

[n,Wp] = ellipord(Wp,Ws,Rp,Rs); % Filter Order

[z,p,k] = ellip(n,Rp,Rs,Wp); % Transfer Function Coefficients

[sos,g] = zp2sos(z,p,k); % Second-Order-Section For Stability

figure(2)

freqz(sos, 2^14, Fs)

yf = filtfilt(sos,g,y);

yu = max(yf);

yl = min(yf);

yr = (yu-yl); % Range of ‘y’

yz = yf-yu+(yr/2);

zt = t(yz(:) .* circshift(yz(:),[1 0]) <= 0); % Find zero-crossings

per = 2*mean(diff(zt)); % Estimate period

ym = mean(y); % Estimate offset

fit = @(b,x) b(1) .* exp(b(2).*x) .* (sin(2*pi*x./(b(3)+b(4).*x) + 2*pi/b(5))) + b(6); % Objective Function to fit

fcn = @(b) sum((fit(b,t) - yf).^2); % Least-Squares cost function

s = fminsearch(fcn, [yr; -1; per; 0.01; -1; ym]) % Minimise Least-Squares

xp = linspace(min(t),max(t));

figure(3)

plot(t,y,'b')

hold on

plot(t,yf,'g', 'LineWidth',2)

plot(xp,fit(s,xp), 'r', 'LineWidth',1.5)

hold off

grid

title('Sample Data 2')

xlabel('Time')

ylabel('Amplitude')

legend('Original Data', 'Lowpass-Filtered Data', 'Fitted Curve')

producing:

##### 5 Comments

Bjorn Gustavsson
on 23 Jan 2020

### More Answers (1)

Bjorn Gustavsson
on 22 Jan 2020

If you plot your data you should be able to see that the zero-crossings are increasingly further apart.

That means that your mode-function will not be able to capture the decay nicely. You might have to introduce another parameter in your function to model this chirp.

HTH

##### 0 Comments

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