Relative degree for rational series

Consider a two letter alphabet $\Sigma = \{a, b\}$. A formal series $f : \Sigma^* \to \mathbb Q$ has relative degree $r \in \mathbb N$ if all words $w \in \Sigma^*$ in its support $f(w) \neq 0$ are of the form $w \in a^r \cdot \Sigma^*$, and $f(a^r \cdot b) \neq 0$. Decide whether a rational series has a relative degree.