  Aldroubi has shown how one can construct any frame $\gtu$ starting with one
frame $\ftu $,using a bounded operator $U$ on $l^2(N)$. We study the
overcompleteness of the frames in terms of properties of $U$. We also discuss
perturbation of frames in the sense that two frames are ``close'' if a certain
operator is compact. In this way we obtain an equivalence relation with the
property that members of the same equivalence class have the same
overcompleteness. On the other hand we show that perturbation in the
Paley-Wiener sense does not have this property. \\ Finally we construct a frame
which is norm-bounded below, but which does not contain a Riesz basis.The
construction is based on the delicate difference between the unconditional
convergence of the frame representation, and the fact that a convergent series
in the frame elements need not converge unconditionally.
