  This paper addresses the important problem of reconstructing a signal from
multiple multirate observations. The observations are modeled as the output of
an analysis bank, and time-domain analysis is carried out to design an optimal
FIR synthesis bank. We pose this as a minimizing the mean-square problem and
prove that at least one optimal solution is always possible. A parametric form
for all optimal solutions is obtained for a non-maximally decimated filter
bank. The necessary and sufficient conditions for an optimal solution, that
results in perfect reconstruction (PR), are derived as time-domain
pseudocirculant conditions. This represents a novel theoretical contribution in
multirate filter bank theory. We explore PR in a more general setting. This
results in the ability to design a synthesis bank with a particular delay in
the reconstruction. Using this delay, one can achieve PR in cases where it
might not have been possible otherwise. Further, we extend the design and
analysis to non-uniform filter banks and carry out simulations to verify the
derived results.
