# Find period of signal $y(t)=\sum_{n=-\infty}^{+\infty}{e^{-|6t+n|}}$

There is a continuous time signal

$$y(t)=\sum_{n=-\infty}^{+\infty}{e^{-|6t+n|}}$$

I want to calculate it's period ($$T$$) however I didn't find any easy way to calculate it. Is there any formula to convert this signal to a periodic signal form? Is it possible to help me?

• Hint: $\sum\limits_{n=-\infty}^\infty a_n=\sum\limits_{n=-\infty}^\infty a_{n+1}$. – metamorphy Oct 11 '19 at 7:22
• so it's period is $\frac{1}{6}$. Is it true? (@metamorphy) – Amin Oct 11 '19 at 9:26

$$a_n=e^{-|6t+n|}$$ Then we can say that: $$\sum_{n=-\infty}^{+\infty}{a_n} = \sum_{n=-\infty}^{+\infty}{a_{n+6T}}$$ This is true for each $$6T \in Z$$. if we replace it in our $$y(t)$$ then we get:
$$y(t)=\sum_{n=-\infty}^{+\infty}{e^{-|6t+n|}}=\sum_{n=-\infty}^{+\infty}{a_n}=\sum_{n=-\infty}^{+\infty}{a_{n+6T}}=\sum_{n=-\infty}^{+\infty}{e^{-|6t+n+6T|}}=\sum_{n=-\infty}^{+\infty}{e^{-|6(t+T)+n|}}=y(t+T)$$
so period of $$y(t)$$ is $$6T \in Z$$ and smallest (main) period is $$T_0=\frac{1}{6}$$