  Assume that $(X, d, \mu)$ is a space of homogeneous type in the sense of
Coifman and Weiss. In this article, motivated by the breakthrough work of P.
Auscher and T. Hyt\"onen on orthonormal bases of regular wavelets on spaces of
homogeneous type, the authors introduce a new kind of approximations of the
identity with exponential decay (for short, $\exp$-ATI). Via such an
$\exp$-ATI, motivated by another creative idea of Y. Han et al. to merge the
aforementioned orthonormal bases of regular wavelets into the frame of the
existed distributional theory on spaces of homogeneous type, the authors
establish the homogeneous continuous/discrete Calder\'on reproducing formulae
on $(X, d, \mu)$, as well as their inhomogeneous counterparts. The novelty of
this article exists in that $d$ is only assumed to be a quasi-metric and the
underlying measure $\mu$ a doubling measure, not necessary to satisfy the
reverse doubling condition. It is well known that Calder\'on reproducing
formulae are the cornerstone to develop analysis and, especially, harmonic
analysis on spaces of homogeneous type.
