  We prove that for matrix algebras $M_n$ there exists a monomorphism $(\prod_n
M_n/\oplus_n M_n)\otimes C(S^1) \to {\cal Q} $ into the Calkin algebra which
induces an isomorphism of the $K_1$-groups. As a consequence we show that every
vector bundle over a classifying space $B\pi$ which can be obtained from an
asymptotic representation of a discrete group $\pi$ can be obtained also from a
representation of the group $\pi\times Z$ into the Calkin algebra. We give also
a generalization of the notion of Fredholm representation and show that
asymptotic representations can be viewed as asymptotic Fredholm
representations.
