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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 $T_g$?

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  • $\begingroup$ Does this help? $\endgroup$
    – Jdip
    Commented Oct 2, 2022 at 16:46
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    $\begingroup$ Not really... I am actually taking Signals and Systems for the first time and not DSP so my basics aren't very strong. $\endgroup$
    – user64710
    Commented Oct 2, 2022 at 17:03
  • $\begingroup$ What you got is fine. The period of the sum is indeed the LCM which in this case 200. $\endgroup$
    – Hilmar
    Commented Oct 2, 2022 at 17:10
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    $\begingroup$ 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? $\endgroup$
    – TimWescott
    Commented Oct 2, 2022 at 18:06

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