# Error when taking the continuous time Fourier transform

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L'O.G. on 11 Jul 2022
Commented: Paul on 11 Jul 2022
I am trying to figure out what the error is associated with taking a Fourier transform. I have a 1D vector A of 130 elements and I know the error associated with each element that is just a number. The error is a 130 element vector called std_A. From that I think I can calculate the error associated with the function that it makes up:
syms t w real
f(t) = sum(exp(-t./A))*heaviside(t);
std_ft(t) = sqrt(sum((abs(t).*exp(-t.*A)*heaviside(t)+dirac(t)).*std_A).^2);
The last line is what I calculated by doing the error propagation for the nonlinear function and is approximate.
Since I think in general you need to take the Fourier transform of the error in the time domain to calculate the error in the frequency domain, I have:
std_fw(w) = fourier(std_ft(t),t,w);
The calculation is very slow I think due to the length of the vector A and also the end result is actually a function of both t and w, which I'm not sure how to work with.
Is there a better way of doing this? I feel like I am making this harder than it needs to be.
L'O.G. on 11 Jul 2022
@Jeffrey Clark I thought f(w) should only be a function of w, but unfortunately the symbolic engine doesn't seem able to do the CTFT completely. See Paul's response below and my comment to him, which is really the crux of my problem.

Paul on 11 Jul 2022
Using some examle data ...
A = 1:5;
std_A = 11:15;
syms t w real
f(t) = sum(exp(-t./A))*heaviside(t)
f(t) = std_ft(t) = sqrt(sum((abs(t).*exp(-t.*A)*heaviside(t)+dirac(t)).*std_A).^2)
std_ft(t) = FWIW, there is no need to use abs(t) here because each is multiplied by heaviside(t).
Anyway, I doubt the signal std_ft(t) has a closed form expression for its Fourier Transform, assuming its Fourier Transform exists.
fourier(std_ft(t),t,w)
ans = Certainly Matlab can't find one, so it just resturns an answer in terms of fourier().
Paul on 11 Jul 2022
Actually, I guess I should have at least run a simple test to see if it works:
syms t w real
A = 1; dtau = 0.1;
f(t) = exp(-t/(A + dtau))*heaviside(t);
F(w) = fourier(f(t),t,w);
f0(t) = exp(-t/A)*heaviside(t);
F0(w) = fourier(f0(t),t,w);
dF(w) = dtau/A^2/(1/A + 1i*w)^2;
figure;
fplot(abs([F(w) , F0(w) , dF(w) , F0(w)+dF(w)]))
legend('F', 'F0', 'dF' , 'F0 + dF') figure;
fplot(angle([F(w) , F0(w) , dF(w) , F0(w)+dF(w)]))
legend('F', 'F0', 'dF' , 'F0 + dF') ### Categories

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