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.
Best Answer
The instantaneous frequency in hertz is $f=\frac{d}{dt}(\nu(t)t)=\nu_0+2at$, so basically that's why the FT extends from $\nu_0$ to $\nu_0+2at_{end}$.
You are thinking of $\nu(t)$ as the frequency, which is incorrect. That's the source of the confusion.