# Actual period of a Periodic Wave represented by Fourier Transform

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)?

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.
• 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.).