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I'm looking for a function whose frequency oscillates around a certain value (say, oscillating between 440 Hz and 880 Hz, at a rate of 1 Hz -- i.e., its frequency goes up and down once per second, preferably in a somewhat linear fashion).

I feel like it should be quite simple, but I'm having trouble coming up with such a function.

I tried $x(t)=\sin(\sin(t)\ t)$, but this doesn't seem to work, because it's not periodic -- I'm looking for something whose frequency goes up and down in a periodic fashion.


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When you ask two such closely related questions, it makes sense to cross-link them so that people don't waste their time answering overlapping parts of them separately. – joriki Nov 24 '11 at 22:55
up vote 2 down vote accepted

This is frequency modulation, widely used for broadcast radio.

As a simple model, something like $t\mapsto A\sin\big(\omega_1( t + \alpha\sin(\omega_2 t))\big)$ should do the trick, where

  • $\omega_1$ is the (angular) frequency of the carrier wave.
  • $\omega_2$ is the (angular) frequency of the signal.
  • $\alpha$ is a modulation depth that should be kept comfortably below $1$.

If the signal is something more complex than a sine wave, replace $\sin(\omega_2 t)$ by an antiderivative of the signal.

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Ahhhh yes indeed, it does. Thanks a lot! (Nice note about the antiderivative, I wouldn't have noticed it!) – Mehrdad Nov 24 '11 at 23:29

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