# How can understand periodicity of a Signal from frequency domain representation?

Is it possible to say a signal is periodic from its frequency domain representation? A periodic signal is sum of its sinus and cosinus. Frequency translation of sinus and cosinus functions are symmetrical impulses. Is it correct to say "a signal is periodic if its frequency domain representation includes symmetrical inpulses"?

A signal is periodic if and only if its frequency spectrum is

• discrete (contains only pulses covering a single locus with a finite area).
• all of those discrete pulses occur at an integral multiple of one frequency (the periodicity).

If you have, for example, $\cos(\pi t)+ \cos(t)$, the resulting spectrum is discrete but the function is not periodic.

• See my comment to Marcus: It's not necessary integer multiple, but their ratio needs to be rational. (In the end this is equal to integer multiple of a fundamental frequency, but with integer multiple it can happen that many many frequency bins are zero). – Maximilian Matthé Mar 29 '17 at 13:20

Is it correct to say "a signal is periodic if its frequency domain representation includes symmetrical inpulses"?

A signal is periodic if, and only if, the Fourier Transform is discrete, and the components are positioned so that the ratios of the resulting frequencies is a rational number. That means it is composed of separate impulses, and there has to be a (hypothetical) raster in which you could put those.

Whether or not these are symmetric doesn't matter.

I've been following your questions a bit and I think that the advice I'd like to give you is the following:

You're obviously following a lecture or a probably relatively good textbook to learn about DSP. That's great!!

Your questions, however, don't show that you try very much to solve your questions yourself by applying the formulas you've learned this far to your problem. However, doing that is the core of DSP – so I'd recommend to do more of a formula-based approach. This is not only because answering formula-less questions is a bit hard at times, but also because it's the skill you'll need to apply DSP knowledge (and to pass an Exam).

• Thank you for your reply and advice. I am new here and not used to write formulas on mobile phone here. Thats why i write my question without formula. My questions related to technical discussions in work group not for passing exam.:) – ylcn Mar 29 '17 at 12:54
• well, all the more important that you're doing sufficient research! I was always very tempted to close your questions as not showing sufficient research of your own. To be honest "I'm on mobile, so no formulas" is not an excuse. Downvoting, – Marcus Müller Mar 29 '17 at 12:56
• To be a bit picky: A time-domain signal is only periodic, if its spectrum is discrete and the ratio between any two non-zero frequencies is a rational number. E.g. if you have a frequency response $X(f)=\delta(f-\sqrt{2})+\delta(f-\sqrt{3})$ this corresponds to $x(t)=\exp(j2\pi\sqrt{2}t)+\exp(j2\pi\sqrt{3}t)$ and I think this signal is not periodic in the common sense (because both frequencies will never align) – Maximilian Matthé Mar 29 '17 at 13:19
• Oh!!! @MaximilianMatthé I was actually missing that! Very, very true! Shoot! Thanks! – Marcus Müller Mar 29 '17 at 14:33