Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use H-q to denote an affine Hecke algebra over k of type (G) over tilde (2) with parameter q. The purpose of this paper is to study representations of H-q by using based rings of two-sided cells of an affine Weyl group W of type (G) over tilde (2). We shall give the classification of irreducible representations of H-q. We also remark that a calculation in  actually shows that Theorem 2 in  needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between H-q and all Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affine Weyl groups, but that is the theme of another paper.