  Signature plays an important role in geometry and topology. In the space with
singularity, Goresky and MacPherson extend the signatures to oriented
pseudomanifolds with only even codimensional stratums by using generalized
Poincare duality of intersection homology. After that Siegel extended the
signature on Witt spaces. Higson and Xie study the $C^*$- higher signature on
Witt space. Followed by the combinatorial framework developed by Higson and
Roe, this paper construct the $C^*$-signature on non Witt space with
noncommutative geometric methods. In conical singular case, we compare
analytical signature of smooth stratified non Witt space by Albin, Leichtnam,
Mazzeo and Piazza.
