In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the crossed product algebra $C(X)\ltimes_{\alpha} G$ which is the vector space $C(X)\otimes\mathbb{C}G$ under the produc $(a\otimes g)(b \otimes h)=a g(b) \otimes gh$, $a,b \in C(X), g,h \in G$.
To understand this better I would like to calculate this in the simplest example I could think of. Let $\mathbb{Z}$ act on $\mathbb{R}$ by translation. Now it should be true that $C(S^1)$ and $C(\mathbb{R})\ltimes \mathbb{Z}$ are Morita equivalent. But how do I find this equivalence? How do I find Morita equivalences between algebra's that should "morally" be the same in general?