Sampling rate conversion systems using a new generalized form of the discrete Fourier transform

A recursive factorization of the polynomial 1-z^N leads to an efficient algorithm for the computation of the discrete Fourier transform (DFT) and the cyclic convolution. The paper introduces a new recursive polynomial factorization of the polynomial when N is highly composite. The factorization is used to define a generalized form of the DFT and to derive an efficient algorithm for the computation. The generalized form of the DFT is shown to be closely related to the polyphase decomposition of a sequence, and is applied for the design of sampling rate conversion systems, it gives not only alternative derivations for the polyphase interpolation and the polyphase decimation by an integer factor, but also a new sampling rate conversion system by a rational factor, which is more efficient than the known rational polyphase implementation when the filter length is large