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How would one go about determining if the following discrete time signal x[n] is periodic, and if it is, determine its fundamental period?

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I understand that the period for the second exponential term is 6, but apart from that I am unable to further my calculation.

The answer to the question above is:

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but I fail to understand how the equation highlighted in yellow comes to be. Any help with this matter would be appreciated.

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The first term is just a constant, so it's not relevant to periodicity. The sequence as written is periodic with a fundamental frequency of 1/6.

Your cited solution makes no sense to me.

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it is more clear if you utilize the Euler's formula $$ \begin{align} x[n+N] &= e^{-2}e^{j(\frac{\pi}{3}(n+N)-\frac{\pi}{4})} \\ &= e^{-2}\Big[ \cos(\frac{\pi}{3}n+\frac{\pi}{3}N-\frac{\pi}{4}) -j \sin(\frac{\pi}{3}n+\frac{\pi}{3}N-\frac{\pi}{4}) \Big] \end{align} $$ Using trigonometric identities ( I will leave it as an exercise to you), $x[n]=x[n+N]$ holds if N=6 which is the smallest value that makes the equality to hold. Indeed, the signal is periodic with a fundamental period $N_o=6$.

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