2 Excerpts

# Real-valued fast Fourier transform algorithms

- 1987

#### 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. 0 I. INTRODUCTION NE of the most important tools in modern digital signal processing applications is the fast Fourier transform (FFT). The FFT efficiently computes the discrete Fourier transform (DFT), a mapping of a length4 complex sequence to its length-N complex spectrum. Although most FFT algorithms are designed to compute the DFT of a complex sequence, in many applications the sequence to be transformed is real-valued. It is widely known that a length-(N / 2) complex FFT algorithm can be used to compute the DFT of a length-N real-valued sequence. It is less well known that algorithms of greater efficiency can be developed by exploiting symmetries within the complex-valued algorithms. Several such modified algorithms have been published, but a general discussion of these techniques is not available in the literature. For this reason, less efficient algorithms are generally used. This paper reviews the general methods for constructing fast algorithms for the computation of the DFT of a real-valued series (RFFT). These methods are applied to all major FFT algorithms, and the resulting algorithms are compared. Inherently real-valued transforms, such as the discrete Hartley transform (DHT), can be used instead of the DFT for power spectrum computation or for fast computation of cyclic convolutions. We compare the perfor

**DOI:**10.1109/TASSP.1987.1165220