# Convolution periodicity

Suppose I have the standard convolution of two functions $x$ and $y$, either continuous convolution or discrete convolution.

If both are periodic, the result of the convolution will also be periodic. But what if only one is periodic? Will the result be periodic under certain conditions?

Assuming $y(t)$ is periodic, a shifted version of $y(t)$, $y(t+T_p) = y(t)$, where $T_p$ is the periodicity.
Convolution is integrating the product of $x(t)$ with a shifted (and flipped) version of $y(t)$. Since the shifted versions repeat after the periodicity $T_p$, the convolution integral result too keeps repeating with the same periodicity.