  An important issue with oversampled FIR analysis filter banks (FBs) is to
determine inverse synthesis FBs, when they exist. Given any complex oversampled
FIR analysis FB, we first provide an algorithm to determine whether there
exists an inverse FIR synthesis system. We also provide a method to ensure the
Hermitian symmetry property on the synthesis side, which is serviceable to
processing real-valued signals. As an invertible analysis scheme corresponds to
a redundant decomposition, there is no unique inverse FB. Given a particular
solution, we parameterize the whole family of inverses through a null space
projection. The resulting reduced parameter set simplifies design procedures,
since the perfect reconstruction constrained optimization problem is recast as
an unconstrained optimization problem. The design of optimized synthesis FBs
based on time or frequency localization criteria is then investigated, using a
simple yet efficient gradient algorithm.
