Topic: Signals and Systems
Which of the three signals given below is periodic?
$x_{1}(t)=2e^{j(t+\frac{\pi}{4})}u(t)$
$x_{2}(n)=u[n]+u[-n]$
$x_{3}(t)=\sum_{k=-\infty}^{\infty}\\{\delta[n-4k]-\delta[n-1-4k]\\}$
- A. $x_{1}(t)$
- B. $x_{2}(n)$
- C. $x_{3}(t)$
- D. None of the above
Answer: C. $x_{3}(t)$
Explanation: $x_1(t)$ is multiplied by the step function $u(t)$, meaning it is zero for $t < 0$, so it is not periodic. $x_2(n) = u[n] + u[-n]$ equals 1 for all $n \neq 0$ and 2 for $n=0$ (or 1 depending on step definition), which is a constant with an anomaly at the origin, thus not periodic. $x_3(n)$ (written as $x_3(t)$ in the question but indexed with n) is a summation of shifted impulses that repeats every 4 samples exactly, making it a periodic sequence with period $N=4$.