Let $K$ be a $p$-adic local field with residue field $k$ such that $[k:k^p]=p^e\<+\infty$ and $V$ be a $p$-adic representation of $\text{Gal}(\overline{K}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K^{\text{pf}}}/K^{\text{pf}})$ where $K^{\text{pf}}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^{\text{pf}}$ containing $k$.