An open problem, namely, how to construct perfect reconstruction filter banks with rational sampling factors, is solved. Such filter banks have N branches, each one having a sampling factor of p i/qi, and their sum equals one. In this way, the well-known theory of filter banks with uniform band splitting is extended to allow for nonuniform divisions of the spectrum. This can be very useful in the analysis of speech and music. The theory relies on two transforms. The first transform leads to uniform filter banks having polyphase components as individual filters. The other results in a uniform filter bank containing shifted versions of same filters. This, in turn, introduces dependencies in design, and is left for future work. As an illustration, several design examples for the (2/3, 1/3) case are given. Filter banks are then classified according to the possible ways in which they can be built. It is shown that some cases cannot be solved even with ideal filters (with real coefficients)