In the present paper we analyze Fréchet frame of the form $\{\varphi(\cdot -j)| j \in \mathbb{Z}^d\}$. With a known condition on $\varphi$, we show that the given sequence constitutes a frame for a test space isomorphic to the space of periodic smooth functions so that its dual is the multiple of the space of periodic distributions by $\hat{\varphi}$.