Question
Is the signal
- $ x(t) = \sum_{k = -\infty}^{\infty} \frac {1}{(t +2k)^2 +1} $
periodic? Answer yes/no and justify your answer mathematically.
Answer
Yes because:
- $ x(t+2) = \sum_{k = -\infty}^{\infty}\frac {1}{(t+2+2k)^2+1} = \sum_{k = -\infty}^{\infty}\frac {1}{(t+2(k+1))^2 + 1} $
Change of variable, let r = k+1
- $ => \sum_{k = -\infty}^{\infty}\frac {1}{(t+2r)^2+1} $
This equation is of the original form, therefore it is periodic.
