  We study sparsity and spectral properties of dual frames of a given finite
frame. We show that any finite frame has a dual with no more than $n^2$
non-vanishing entries, where $n$ denotes the ambient dimension, and that for
most frames no sparser dual is possible. Moreover, we derive an expression for
the exact sparsity level of the sparsest dual for any given finite frame using
a generalized notion of spark. We then study the spectral properties of dual
frames in terms of singular values of the synthesis operator. We provide a
complete characterization for which spectral patterns of dual frames are
possible for a fixed frame. For many cases, we provide simple explicit
constructions for dual frames with a given spectrum, in particular, if the
constraint on the dual is that it be tight.
