Here is my signal


cos(n/2) has period 4pi and cos(pi*n/4) has period 8

Now, the question is will the signal be periodic for fundamental period 32pi ?

  • $\begingroup$ If $n$ is a discrete time variable (i.e. it assumes only integer values), then $\cos(n/2)$ is not a periodic signal, simply because $4\pi$ isn't rational. $\endgroup$
    – Matt L.
    Jul 29 '14 at 10:17
  • $\begingroup$ @Matt-I yeah, that is where I was confused. So, the conclusion should be that it is periodic if it is analog and aperiodic if it is digital right ? $\endgroup$ Jul 29 '14 at 10:19
  • 1
    $\begingroup$ In discrete time it is indeed not periodic. $\endgroup$
    – Matt L.
    Jul 29 '14 at 10:26


  • Now finding out period of $\cos\left(\dfrac{n}{2}\right)$

    $\cos\left(\dfrac{n}{2}\right)=\cos\left(2*\pi*\dfrac{n}{N1}\right) \implies \dfrac{1}{2}=2*\dfrac{\pi}{N1} \implies N1=4*\pi$

  • Now finding out period of second function $\cos\left(\pi*\dfrac{n}{4}\right)$

    $\cos\left(\pi*\dfrac{n}{4}\right) = \cos\left(2*\pi*\dfrac{n}{N2}\right)\implies \dfrac{\pi}{4}=2*\dfrac{\pi}{N2} \implies N2=8$.

Now for total function period is LCM of $(N1,N2)$ i.e, $\mathrm{LCM}(4*\pi,8) \implies32*\pi$.

Therefore, the period of the function is $\boxed{32*\pi}$.


You can try using this:

(1) cos(a+b) = cos(a)*cos(b)+sin(a)*sin(b)
(2) cos(a-b) = cos(a)*cos(b)-sin(a)*sin(b)

Then you can do (1)+(2) and get that


Then you can identify the period of cos(a+b)-cos(a-b) by using this for example http://fourier.eng.hmc.edu/e101/lectures/Fundamental_Frequency.pdf

Hope this helps

  • $\begingroup$ $\cos(a)\cos(b)=\frac12[\cos(a+b)+\cos(a-b)]$. Please try for yourself to find the period, and then you might want to reconsider your answer. $\endgroup$
    – Matt L.
    Jul 29 '14 at 10:24
  • $\begingroup$ summation or multiplication doesn't make difference in calculation of period. $\endgroup$ Jul 29 '14 at 10:30
  • $\begingroup$ Sorry, I mixed them up... Should have looked them up before writing. I also think I missed the point of the question, the answer is more a general approach to finding periods of multiplied sinusoid signals... $\endgroup$
    – schvaba986
    Jul 29 '14 at 11:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.