Data-canonic real FFT flow-graphs for composite lengths

This paper presents a novel algorithm to compute real-valued fast Fourier transform (RFFT) that is canonic with respect to the number of signal values. A signal value can correspond to a purely real or purely imaginary value, while a complex signal consists of 2 signal values. For an N-point RFFT, each stage need not compute more than N signal values, since the degrees of freedom of the input data is N. Any more than N signal values computed at any stage is inherently redundant. Canonic RFFT algorithms are defined as those RFFT algorithms that have the least number of signals at each FFT stage. In order to reduce the redundant samples, a sample removal lemma, and two types of twiddle factor transformations are proposed: pushing and modulation. We consider 4 different cases. Canonic RFFT for any composite length can be computed by applying the proposed algorithm recursively. Performances of different RFFTs are also discussed in this paper. The major advantages of the canonic RFFTs are that they require the least butterfly operations, lead to more regular sub-blocks in the data-flow, and only involve real datapath when mapped to architectures.