There holds the global Atiyah-Singer Index Theorem for general elliptical differential operators. But for local index theorem,people can only give a proof for concrete operator and the teclnique used in the proof varies from one to another.We present here the suitable definition of de Rham-Hodge-Signature operator D and the proof of its locaf index theorem.From[6],we conclude that the local index for D is neither of type α nor type β. This is dtherent from the case of classical elliptic differential operators.