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For instance, I have signals such as

$$x(t)=e^{\cos(t)}$$

$$x(t)=t\cdot e^{\cos(t)}$$

How should I approach these kinds of signals to determine their fundamental periods?

The fundamental period is the smallest $p > 0$ for which $x(t) = x(t+p)$ holds.

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  • $\begingroup$ Can you add the definition you are using for "fundamental period" to the question? $\endgroup$
    – The Count
    Jan 28, 2017 at 19:05
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    $\begingroup$ Of course, I edited it $\endgroup$
    – Aldrich Taylor
    Jan 28, 2017 at 19:06
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    $\begingroup$ The fundamental period of $x$ is $p$ such $x(t)=x(t+p)$ where p is minimal. So for the first function $2 \pi$ & the second function is not periodic. $\endgroup$
    – Donald Splutterwit
    Jan 28, 2017 at 19:09

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For the fist $x(t)$, know that the exponential is a bijective function on the real line, so the period of $x(t)$ is the same as the period of $\cos(t)$, namely $2\pi$. For the second $x(t)$, there is no fundamental period because the function is not periodic. Any continuous periodic function must be bounded, but clearly the second choice of $x(t)$ is not.

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  • $\begingroup$ Are you saying that $2\pi$ is not a real number? Because it is. $\endgroup$
    – Paul
    Jan 28, 2017 at 20:55
  • $\begingroup$ @Paul, I never claimed that $2\pi$ is not a real number. I claimed that the function $x(t) = t e^{\cos(t)}$ is not periodic. You can check that $x(t + 2\pi) = (t+2\pi)e^{\cos(t)} = x(t) + 2\pi e^{\cos(t)} \neq x(t)$. $\endgroup$
    – nullUser
    Jan 29, 2017 at 16:53
  • $\begingroup$ What I said was a reply to a comment that is now deleted $\endgroup$
    – Paul
    Jan 29, 2017 at 17:43

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