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Suppose there is a Periodic Signal with period 2π with no discontinuity. We can represent this wave using infinite sum of scaled version of sines and cosines of different frequencies (Fourier Transform). So what do we say about the frequency of our wave, 2π or a mixture of different frequencies involved in its Fourier Transform?

I want to know this to know the output of an LTI system which acts as a filter and removes signals with frequency less than π, so if I pass my wave then will it be affected (as it has waves with frequency less than π) or it won't(as the whole signal has frequency 2π)?

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By definition, the period $P$ of a periodic signal $s$, sometimes called fundamental period, is the smallest $P>0$ such that $s (t+P) = s (t)$ for all $t$. The (fundamental) frequency is then $f=1/P$.

If the fundamental period is $2\pi$ (i.e. the fundamental frequency is $1/2\pi$) and if there are spectral components with period $2\pi/2$, $2\pi/3$, etc. (i.e. frequency $2/2\pi$, $3/2\pi$, etc.), then a low-pass filter which modifies the spectral content at periods smaller than $\pi$ will modify the signal. A high-pass filter which modifies the spectral content at periods larger than $\pi$ will modify the signal too.

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Generally, we would be talking about DSP. There's two easy ways to LTI filter signals in this context:

  • FIR filters - those are usually "windowed" to smooth out the frequency response. This way, there's a gentle, well defined slope from "pass" to "stop" (and just finitely many frequencies with response amplitude $0$ etc.).
  • IIR filters - those are inherently smooth, however it might still be hard to define "the frequency" of a given signal. Again, apply windowing before the FFT and look for a maximum.

The basic principle is Heisenberg uncertainty. In the continuous case, this extends to "the only signal with a well-defined frequency is an eternal sine wave". This is not very practical.

The way you stated the question shows that you haven't worked much in this area.

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