I'm working on an oscillating signal whose trend can be modelled as a frequency linearly varying function. An example may be as follows: $ \Gamma(t)=\sin(2\pi\nu(t)t) $ with $ \nu(t)=\nu_0 + at $ My signal is defined in a time interval as the following: $ t=[0,t_\mathrm{end}] $
When I Fourier Transform $\Gamma(t)$ getting $\Phi(\nu)$ ($\Phi(\nu)=FT[\Gamma(t)]$), I expect in the frequency domain a large peak extending from $\nu_0$ to $\nu_0 + at_\mathrm{end}$. Instead, what I obtain is a large peak extending from $\nu_0$ to $\nu_0 + 2at_\mathrm{end}$, centered at $\nu_0 + at_\mathrm{end}$.
Is this a feature of the Fourier Transform? I cannot understand what's going on.
Thank you very much.