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Split-radix FFT algorithm
About: Split-radix FFT algorithm is a research topic. Over the lifetime, 1845 publications have been published within this topic receiving 41398 citations.
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19 Apr 1994
TL;DR: An algorithm is developed, called the quick Fourier transform (QFT), that will reduce the number of floating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths.
Abstract: This paper will look at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of discrete Fourier transform (DFT). We will develop an algorithm, called the quick Fourier transform (QFT), that will reduce the number of floating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths. Further by applying the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(N log N) algorithm. The algorithm can be easily modified to compute the DFT with only a subset of input points, and it will significantly reduce the number of operations when the data are real. The simple structure of the algorithm and the fact that it is well suited for DFTs on real data should lead to efficient implementations and to a wide range of applications. >
26 citations
TL;DR: A fast algorithm for the evaluation of the Fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast Fouriertransform (FFT) algorithm is presented.
Abstract: In computational electromagnetics and other areas of computational science and engineering, Fourier transforms of discontinuous functions are often required. We present a fast algorithm for the evaluation of the Fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast Fourier transform (FFT) algorithm. We call this the discontinuous FFT algorithm. For N sample points, the complexity of the algorithm is O(/spl nu/Np+/spl nu/Nlog(N)) where p is the interpolation order and /spl nu/ is the oversampling factor. The method also provides a new nonuniform FFT algorithm for continuous functions. Numerical experiments demonstrate the high efficiency and accuracy of this discontinuous FFT algorithm.
26 citations
TL;DR: This paper shows how a similar approach can be used for sequences which are known to have only odd harmonics, and is shown to be essentially the dual of the known method for time symmetry.
Abstract: It is well known that if a finite duration, N-point sequence x(n) possesses certain symmetries, the computation of its discrete Fourier transform (DFT) can be obtained from an FFT of size N/2 or smaller. This is accomplished by first preprocessing the sequence, taking the FFT of the processed sequence, and then postprocessing the results to give the desired transform. In this paper we show how a similar approach can be used for sequences which are known to have only odd harmonics. The approach is shown to be essentially the dual of the known method for time symmetry. Computer programs are included for implementing the special procedures discussed in this paper.
26 citations
TL;DR: Among various discrete transforms, discrete Fourier transformation (DFT) is the most important technique that performs Fourier analysis in various practical applications, such as digital signal processing, wireless communications, to name a few.
Abstract: Discrete Fourier transform (DFT) is an important transformation technique in signal processing tasks. Due to its ultrahigh computing complexity as $O(\!N^{\!2}\!)$ , $N$ - point DFT is usually implemented in the format of fast Fourier transformation (FFT) with the complexity of $O(N\log N)$ . Despite this significant reduction in complexity, the hardware cost of the multiplication-intensive $N$ - point FFT is still very prohibitive, particularly for many large-scale applications that require large $N$ . This brief, for the first time , proposes high-accuracy low-complexity scaling-free stochastic DFT/FFT designs. With the use of the stochastic computing technique, the hardware complexity of the DFT/FFT designs is significantly reduced. More importantly, this brief presents the scaling-free stochastic adder and the random number generator sharing scheme, which enable a significant reduction in accuracy loss and hardware cost. Analysis results show that the proposed stochastic DFT/FFT designs achieve much better hardware performance and accuracy performance than state-of-the-art stochastic design.
26 citations
TL;DR: Presents a very short, simple, easy to understand bit-reversal algorithm for radix-2 fast Fourier transform (FFT), which is, furthermore, easily extendable to Radix-M and Yong's technique, which is comparable to that of the fastest algorithms.
Abstract: Presents a very short, simple, easy to understand bit-reversal algorithm for radix-2 fast Fourier transform (FFT), which is, furthermore, easily extendable to radix-M. In addition, when implemented together with Yong's (see IEEE Trans. Acoust., Speech, Signal Processing, vol.39, no.1O, p.2365-7, 1991) technique, the computing time is comparable to that of the fastest algorithms. >
26 citations