Prime factor FFT algorithms for real-valued series

Abstract

This paper presents two techniques for computing a discrete transform of a vector of real-valued data using the Prime Factor Algorithm (PFA) with high-speed convolution. These techniques are applied to the Discrete Fourier Transform (DFT) and the Discrete Hartley Transform (DHT). The primary goals of these techniques are to eliminate unnecessary computations required when implementing a complex transform on a real-valued vector, to compute the transform in-place in the original length-N real vector, and to obtain the transform coefficients in-order. The two algorithms described require modification of the Winograd short-length transform modules to accommodate a real input. One technique replaces the modules in the Burrus-Eschenbacher PFA program with the modified real-input modules and constructs the complete transform in a final step of additions and subtractions after modules for each factor have been executed. The other technique uses these real-input DFT modules for part of the computation associated with each factor and requires complex input DFT modules for another part of the computation. These algorithms require exactly one half of the number of multiplications and slightly less than one half of the number of additions required by a complex-input PFA.