Generalized sampling and the perfect reconstruction problem for maximally decimated filter banks

A digital signal {x(n)} is used as a common input to a parallel bank of m digital linear shift-invariant (LSI) analysis filters. The m filter outputs are then both downsampled (decimated) by a factor of m and unsampled (interpolated) by the same factor m in each channel. The perfect reconstruction problem is that of finding digital synthesis filters in each channel such that an additive combination of these filter outputs results in restoration of the original sequence. It is shown that a solution of the perfect reconstruction problem in the digital domain can be obtained from the solution of a corresponding analog generalized sampling problem using a bandlimited input (which interpolates the discrete input at integer points) and analog filters that are simply related to the given digital filters of the maximally decimated filter bank