  In this paper we give a description of the depth-$r$ Bernstein center for
non-negative integers $r$ of a reductive simply connected group $G$ over a
non-archimedean local field as a limit of depth-$r$ standard parahoric Hecke
algebras. Using the description, we construct maps from the algebra of stable
functions on the $r$-th Moy-Prasad filtration quotient of hyperspecial
parahorics to the depth-$r$ Bernstein center and use them to attach to each
depth-$r$ irreducible representation $\pi$ an invariant $\theta(\pi)$, called
the depth-$r$ Deligne-Lusztig parameter of $\pi$. We show that $\theta(\pi)$ is
equal to the semi-simple part of minimal $K$-types of $\pi$.
