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I never got a chance to study Fourier Transforms in school. What I do know about them (which isn't much) I have gleaned from the web. I understand that you can transform any periodic signal between the time domain and the frequency domain. So here's my question:

I have a periodic signal in the time domain:

$$f(t) = \\ \frac{6 (Sin[t] + 5/16 Sin[3 t] + 3/20 Sin[5 t] + 1/16 Sin[7 t])}{\pi} \\ + \frac{4 (Sin[2 t] + 5/16 Sin[6 t] + 3/20 Sin[10 t] + 1/16 Sin[14 t])}{\pi} \\ - 3/2$$

If you graph this function you can see it oscillates between a max value of about 1 to a min value of about -4.

Is there a way I can use a Fourier Transform to reduce or limit the amplitude of this signal between -1 and 1 ?

In my mind the steps of this may look something like:

  • Step 1: Transform time-based signal f(t) to frequency-based signal $g(f)$.
  • Step 2: apply some function to $g(f)$
  • Step 3: use inverse transform of $g(f)$ (that has had limiter function applied) to get $f'(t)$

Now all $f'(t)$ values are between -1 and 1 (for example).

Thanks a lot for any education/enlightenment you could pass along.

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    $\begingroup$ But why not just multiply the function in the time domain by some function that limits the amplitude e.g 1/4? $\endgroup$ – Graham Hesketh Mar 26 '14 at 20:11
  • $\begingroup$ You could use a filter in the frequency domain to cut off some harmonics ($\sin(m\phi)$ terms) if that is what you are thinking of but that wont directly correspond to a specific effect on the amplitude... $\endgroup$ – Graham Hesketh Mar 26 '14 at 20:13
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That is not the way to use a Fourier transform. To influence the amplitude of the signal in the time domain, it is best to transform it in the time domain as Graham already indicated.

Ah well, if you really want to, you can do it. By increasing the frequency at 0 Hz you will effectively lift the signal up. And by decreasing the amplitude of all frequencies, you will effectively also reduce all amplitudes in the time domain.

But really, in the frequency domain you're supposed to influence the frequencies. You can apply a low pass, a high pass, or a band filter to eliminate unwanted frequencies. Suppose you have 50/60 Hz noise, which is typically generated by the global electromagnetic field of standard current, you can use the Fourier transform with a band filter to filter it out. After transforming it back to the time domain, the noise will be gone! :)

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