  We give a parametrix construction for the signature operator on any compact,
oriented, stratified pseudomanifold X which satisfies the Witt condition. This
construction is inductive. It is then used to show that the signature operator
is essentially self-adjoint and has discrete spectrum of finite multiplicity,
so that its index -- the analytic signature of X -- is well-defined. We then
show how to couple this construction to a C^*_r(Gamma) Mischenko bundle
associated to any Galois covering of X with covering group Gamma. The
appropriate analogues of these same results are then proved, and it follows
that we may define an analytic signature class as an element of the K-theory of
C^*_r(Gamma). In a sequel to this paper we establish in this setting the full
range of conclusions for this class which sometimes goes by the name of the
signature package.
