Say I have a 1D (spatial) signal (resolution = $1000$) which is zero everywhere except from $x = 250$ to $750$, where it equals one.
I ultimately want to calculate the spatial width of this signal using FFTs. Of course we know the width here to be $500$; in actuality, I am dealing with a signal that evolves with time and wish to calculate the average "pulse" width over all the time frames, so I do not know the widths. I have opted to use FFTs in this pursuit, so I must conduct a "sanity check" to make sure the method works. This method was suggested to me by a colleague whose intuition is many leagues farther than my own, so if someone could explain the intuition to me, I would appreciate it a lot.
Step 1: Subtract the DC background (subtract the mean from every point of the signal).
Step 2: Take the FFT of the signal, then the power (the Fourier transform times the complex conjugate of it). Normalize the power spectrum.
Step 3: Calculate the half-width at half-maximum (HWHM); here half-width is the half-width of the peak in k-space, of course.
Step 4: Convert this k-space HWHM back to real-space: real-space width = 1 / (HWHM / resolution).
When I do these steps for the signal above, I calculate a real-space width of $1189427$, laughably off from $500$. Where does the method go wrong?