Oversampled Paraunitary DFT Filter Banks: A General Construction Algorithm and Some Specific Solutions

Oversampled filter banks are at the center of essential signal processing systems, either for source coding or for transmission applications. Among this class of filter banks (FBs), the case of oversampled discrete Fourier transform (DFT) filter banks is particularly important since it leads to efficient algorithm implementations. However, up to now, no general construction algorithm has been proposed to exhaustively cover the large set of the various solutions offered by this family of FBs. The paper recalls in detail the precise features of the rectangular paraunitary matrices that are specific of this type of systems. A parametrization method is proposed involving transformations, being either rotations or shifts, that depend upon the oversampling ratio and upon the prototype filter length. This shows that several factorizations exist leading to solutions of different dimension. Orthogonal and paraunitary patterns are introduced and, using a sequence of transformations applied to a pattern (STAP), an algorithm is derived that allows us to get the exhaustive set of solutions for various values of the oversampling ratio and prototype filter length. Furthermore, for some oversampling ratios, explicit expressions of some solutions are also provided that are valid whatever the maximum length of the prototype filter. Finally, design examples for linear and nonlinear phase prototype filters are presented.