3
$\begingroup$

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.

  • 2
    Engineers call this kind of signal a _chirp signal_. You might want to try your question on dsp.SE where a fair number of participants are quite familiar with these signals.2012-05-15
  • 0
    Yes, a sinuosoid with an "intantaneous frequency" that varies continuosly between $f_1$ and $f_2$ (FM) would seem to have a Fourier transform with support (non zero values) in the $f_1 f_2$. But things are more complicated, FM introduces harmonics and hence to have larger frencuencies is to be expected. http://en.wikipedia.org/wiki/Chirp#Linear_chirp2012-05-15
  • 0
    I expected it to be much more trivial than it was.. thanks a lot for the link!2012-05-15

2 Answers 2