# What would be the fundamental period of this discrete-time signal?

Question: $$x[n]= 3\cos(0.3\pi n + 0.1) - \sin(0.11\pi n -\frac{\pi}{3})$$ Find the fundamental period for this discrete signal.

My attempt: Write the two functions as:

$$\underbrace {3\cos\left(\frac{3}{10}\pi n + 0.1\right)}_{=f(n)} - \underbrace{\sin\left(\frac{11}{100}\pi n -\frac{\pi}{3}\right)}_{=g(n)}$$

The period of $$f(n)$$ is as follows:

$$T_f=\frac{2\pi}{\frac{3\pi}{10}}\cdot k$$ For $$k=3$$, $$T_f=20$$

The period of $$g(n)$$ is as follows:

$$T_g=\frac{2\pi}{\frac{11\pi}{100}}\cdot m$$ For $$m=11$$, $$T_g=200$$

Now since both of these discrete-time signals are periodic there sum should also be periodic but with what fundamental period?

Would it just be the $$\textbf{LCM}$$ of $$T_f$$ and $$T_g$$?

• Does this help?
– Jdip
Oct 2, 2022 at 16:46
• Not really... I am actually taking Signals and Systems for the first time and not DSP so my basics aren't very strong.
– user64710
Oct 2, 2022 at 17:03
• What you got is fine. The period of the sum is indeed the LCM which in this case 200. Oct 2, 2022 at 17:10
• Well, you've got something that repeats every 20 time intervals -- which means it repeats every 40, 60, etc. time intervals. Then you have something else that repeats every 200 (and 400 and 600, etc.) time intervals. Can you reason out a common interval where both will repeat? Oct 2, 2022 at 18:06