Splitting a signal into N filtered channels subsampled by N is an important problem in digital signal processing. A fundamental property of such a system is that the original signal can be perfectly recovered from the subsampled channels. It is shown that this can always be done, and that FIR solutions exist. This is done by mapping the NM-dimensional nonlinear problem (where N is the number of channels and M the length of the FIR filters) into an M-dimensional linear problem. For N = 2, a general class of FIR solutions is derived, together with methods to find filters. The dual problem of mixing N signals into one channel upsampled by N is also addressed. Several applications are proposed. All results are obtained by looking at the N filter bank as a true N channel system, rather than N separate channels.