Real-valued fast Fourier transform algorithms

Abstract

This tutorial paper describes the methods for constructing fast algorithms for the computation of the discrete Fourier transform (DFT) of a real-valued series. The application of these ideas to all the major fast Fourier transform (FFT) algorithms is discussed, and the various algorithms are compared. We present a new implementation of the real-valued split-radix FFT, an algorithm that uses fewer operations than any other real-valued power-of-2-length FFT. We also compare the performance of inherently real-valued transform algorithms such as the fast Hartley transform (FHT) and the fast cosine transform (FCT) to real-valued FFT algorithms for the computation of power spectra and cyclic convolutions. Comparisons of these techniques reveal that the alternative techniques always require more additions than a method based on a real-valued FFT algorithm and result in computer code of equal or greater length and complexity.

A New Efficient FDTD Time-to-Frequency-Domain Conversion Algorithm

“…Utilizing this information can halve the computational cost of the FFT of (8). A number of techniques for efficiently computing the FFT of a real-valued sequence have been presented by Sorensen et al in [11] and other researchers. Among them, a simple technique is to use the symmetry of the FFT to transform two real-valued sequences simultaneously by computing one complex-FFT.…”

“…Thus, transforming two real-valued sequences costs only the same as one complex-FFT. More details about this technique can be found in [11]. We incorporate this technique into the modified NUFFT algorithm for reducing the computational cost of (8).…”

“…First, we modify the original NUFFT of [8] to have real interpolation coefficients, which halves the computational cost of the interpolation processing. Second, we utilize a real-valued FFT technique [11] to reduce the oversampling FFT computation required for the NUFFT implementation. Third, we develop a segmentation strategy to remove the requirement of storing the complete time histories of FDTD sequences.…”

An analysis of real-Fourier domain-based adaptive algorithms implemented with the Hartley transform using cosine-sine symmetries

“…In addition, since we have assumed real inputs and weights, it would be unfair to compare a complex FFT with a real DCT, and hence, in our analysis, we use specialized FFT algorithms that work with real data [23]. However, our choice for the real algorithms is restricted to the radix-2 counterparts on both sides since our motive is in showing the relative advantages rather than comparing the individual algorithms.…”

High-speed area-efficient recursive DFT/IDFT architectures

“…Without loss of generality, the input data is assumed as complex-valued data. From existing research, there are possible four categories for the structures of DFT computations: 1) recursive-algorithm based architecture [1][2][3][4][5][6], 2) butterfly-based architecture [1,7], 3) ROM operation based structure [8], and 4) multiplier-accumulator based structure. It is well known that the DFT architectures based on the recursive algorithm are more area-efficient than those realized by other approaches.…”