# Determine whether a signal is periodic or not and get fundamental period

Here is my signal

Cos(n/2)*cos(pi*n/4)


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 ?

• 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. – Matt L. Jul 29 '14 at 10:17
• @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 ? – Sujit Maharjan Jul 29 '14 at 10:19
• In discrete time it is indeed not periodic. – Matt L. Jul 29 '14 at 10:26

$$\cos\left(\dfrac{n}{2}\right)*\cos\left(\pi*\dfrac{n}{4}\right)$$

• 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

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


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

• $\cos(a)\cos(b)=\frac12[\cos(a+b)+\cos(a-b)]$. Please try for yourself to find the period, and then you might want to reconsider your answer. – Matt L. Jul 29 '14 at 10:24
• summation or multiplication doesn't make difference in calculation of period. – Sujit Maharjan Jul 29 '14 at 10:30
• 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... – schvaba986 Jul 29 '14 at 11:05