what is the periodicity of unit impulse $\delta[n-2m-1]$?

Question

I want to find the periodicity of the following function: $\delta[n-2m-1]$ I have calculated the periodicity of the above functions which is $2$ ,as $m=0,1,2.... \delta[n-1],\delta[n-3],\delta[n-5]$.

Answer 1

$\delta[n-2m-1]$ for a single $m$ is not periodic. I think you meant whether $$y[n]=\sum_{m=-\infty}^{+\infty}\delta[n-2m-1]$$ is periodic with period $2$.

The answer is yes, since

$$\begin{align}y[n+2]&=\sum_{m=-\infty}^{+\infty}\delta[n+2-2m-1]\\ &=\sum_{m=-\infty}^{+\infty}\delta[n-2(m-1)-1]\\ &=\sum_{m'=-\infty}^{+\infty}\delta[n-2m'-1]\\ &=y[n]\end{align}$$