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.

Fast numerical algorithm for the linear canonical transform.

Abstract: The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an NlogN algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

Integer fast Fourier transform

Abstract: A concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier transform is introduced. Unlike the fixed-point fast Fourier transform (FxpFFT), the new transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the transform, is presented and confirmed by a simulation. The transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.

FFT algorithms for vector computers

Abstract: The adaptation of the Cooley-Tukey, the Pease and the Stockham FFT's to vector computers is discussed. Each of these algorithms computes the same result namely, the discrete Fourier transform. They differ only in the way that intermediate computations are stored. Yet it is this difference that makes one or the other more appropriate depending on the application. This difference also influences the computational efficiency on a vector computer and motivates the development of methods to improve efficiency. Each of the FFT's is defined rigorously by a short expository FORTRAN program which provides the basis for discussions about vectorization. Several methods for lengthening vectors are discussed, including the case of multiple and multi-dimensional transforms where M sequences of length N can be transformed as a single sequence of length MN using a 'truncated' FFT. The implementation of an in place FFT on a computer with memory-to-memory architecture is made possible by in place matrix-vector multiplication.

Vectorizing the FFTs

Abstract: Publisher Summary This chapter provides an overview on vectorizing the FFTs. The fast Fourier transform (FFT) is the most well known of all algorithms. It is superior to the slow transform and has applications in all areas of scientific computing. The term FFT was applied to a specific algorithm for the rapid computation of the discrete complex Fourier transform; however, it has become a generic term that is applied to any one of a large number of algorithms that compute the complex as well as other Fourier transforms. Many algorithms exist for a given Fourier transform, and when they are applied to a particular sequence, the result is the same. However, the algorithms differ in the ways that intermediate results are computed and stored. It is these important differences that provide the algorithms with unique properties that make one or the other more attractive for a particular application.

Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms

Abstract: PRELIMINARIES An Elementary Introduction to the Discrete Fourier Transform Some Mathematical and Computational Preliminaries SEQUENTIAL FFT ALGORITHMS The Divide-and-Conquer Paradigm and Two Basic FFT Algorithms Deciphering the Scrambled Output from In-Place FFT Computation Bit-Reversed Input to the Radix-2 DIF FFT Performing Bit-Reversal by Repeated Permutation of Intermediate Results An In-Place Radix-2 DIT FFT for Input in Natural Order An In-Place Radix-2 DIT FFT for Input in Bit-Reversed Order An Ordered Radix-2 DIT FFT Ordering Algorithms and Computer Implementation of Radix-2 FFTs The Radix-4 and the Class of Radix-2s FFTs The Mixed-Radix and Split-Radix FFTs FFTs for Arbitrary N FFTs for Real Input FFTs for Composite N Selected FFT Applications PARALLEL FFT ALGORITHMS Parallelizing the FFTs: Preliminaries on Data Mapping Computing and Communications on Distributed-Memory Multiprocessors Parallel FFTs without Inter-Processor Permutations Parallel FFTs with Inter-Processor Permutations A Potpourri of Variations on Parallel FFTs Further Improvement and a Generalization of Parallel FFTs Parallelizing Two-Dimensional FFTs Computing and Distributing Twiddle Factors in the Parallel FFTs APPENDICES Fundamental Concepts of Efficient Scientific Computation Solving Recurrence Equations by Substitution Bibliography