To do a nonlinear fit in the frequency domain, you will need to calculate a function that is the Fourier transform of the time-domain function you want to fit.
Try this:
syms k1 k2 t Tmax w w0
y(t,w0,k1,k2) = exp(k1*t) * cos(k2*w0*t)
Y = int(y*exp(1j*w*t), t, 0, Tmax)
Y = simplify(Y, 'Steps',250)
Yf = matlabFunction(Y) % Individual Parameters
Yf = matlabFunction(Y, 'Vars',{[w0,k,Tmax],w}) % Parameter Vector ‘in1’
producing:
Yf = @(in1,w) -(in1(:,2)+w.*1i)./((in1(:,2)+w.*1i).^2+in1(:,1).^2)+(exp(in1(:,3).*in1(:,2)+in1(:,3).*w.*1i).*(in1(:,1).*sin(in1(:,3).*in1(:,1))+cos(in1(:,3).*in1(:,1)).*(in1(:,2)+w.*1i)))./((in1(:,2)+w.*1i).^2+in1(:,1).^2);
where ‘k’ (‘in(:,2)’) is the exponential decay constant, ‘w0’ (‘in(:,1)’) is the frequency of the cosine function, and ‘Tmax’ (‘in(:,3)’)is the maximum (end) time of the vector, and ‘w’ is the independent variable and radian frequency. Note that this produces a complex result, so it might be easiest to take the abs() of it and use it with the abs() of the Fourier transform. Also, as written here, ‘in’ is defined as a row vector, and will throw an error if you use a column vector as your initial parmeter estimates.
I have never done any nonlinear parameter estimation in the frequency domain, so I have no experience with it. I also did not test this function with the Fourier transform of a decaying cosine function, so I have no idea if it will succeed.
Have fun!
