  Let $L/K$ be an extension of complete discrete valuation fields, and assume
that the residue field of $K$ is perfect and of positive characteristic. The
residue field of $L$ is not assumed to be perfect.
  In this paper, we prove a formula for the Swan conductor of the image of a
character $\chi \in H^1(K, \mathbb{Q}/\mathbb{Z})$ in $H^1(L,
\mathbb{Q}/\mathbb{Z})$ for $\chi$ sufficiently ramified. Further, we define
generalizations $\psi_{L/K}^{\mathrm{ab}}$ and $\psi_{L/K}^{\mathrm{AS}}$ of
the classical Hasse-Herbrand $\psi$-function and prove a formula for
$\psi_{L/K}^{\mathrm{ab}}(t)$ for sufficiently large $t\in \mathbb{R}$.
