Cooley–Tukey FFT algorithm

Abstract: 1 Introduction.- 1.1 Introductory Remarks.- 1.2 Notations.- 1.3 The Structure of the Book.- 2 Elements of Number Theory and Polynomial Algebra.- 2.1 Elementary Number Theory.- 2.1.1 Divisibility of Integers.- 2.1.2 Congruences and Residues.- 2.1.3 Primitive Roots.- 2.1.4 Quadratic Residues.- 2.1.5 Mersenne and Fermat Numbers.- 2.2 Polynomial Algebra.- 2.2.1 Groups.- 2.2.2 Rings and Fields.- 2.2.3 Residue Polynomials.- 2.2.4 Convolution and Polynomial Product Algorithms in Polynomial Algebra.- 3 Fast Convolution Algorithms.- 3.1 Digital Filtering Using Cyclic Convolutions.- 3.1.1 Overlap-Add Algorithm.- 3.1.2 Overlap-Save Algorithm.- 3.2 Computation of Short Convolutions and Polynomial Products.- 3.2.1 Computation of Short Convolutions by the Chinese Remainder Theorem.- 3.2.2 Multiplications Modulo Cyclotomic Polynomials.- 3.2.3 Matrix Exchange Algorithm.- 3.3 Computation of Large Convolutions by Nesting of Small Convolutions.- 3.3.1 The Agarwal-Cooley Algorithm.- 3.3.2 The Split Nesting Algorithm.- 3.3.3 Complex Convolutions.- 3.3.4 Optimum Block Length for Digital Filters.- 3.4 Digital Filtering by Multidimensional Techniques.- 3.5 Computation of Convolutions by Recursive Nesting of Polynomials.- 3.6 Distributed Arithmetic.- 3.7 Short Convolution and Polynomial Product Algorithms.- 3.7.1 Short Circular Convolution Algorithms.- 3.7.2 Short Polynomial Product Algorithms.- 3.7.3 Short Aperiodic Convolution Algorithms.- 4 The Fast Fourier Transform.- 4.1 The Discrete Fourier Transform.- 4.1.1 Properties of the DFT.- 4.1.2 DFTs of Real Sequences.- 4.1.3 DFTs of Odd and Even Sequences.- 4.2 The Fast Fourier Transform Algorithm.- 4.2.1 The Radix-2 FFT Algorithm.- 4.2.2 The Radix-4 FFT Algorithm.- 4.2.3 Implementation of FFT Algorithms.- 4.2.4 Quantization Effects in the FFT.- 4.3 The Rader-Brenner FFT.- 4.4 Multidimensional FFTs.- 4.5 The Bruun Algorithm.- 4.6 FFT Computation of Convolutions.- 5 Linear Filtering Computation of Discrete Fourier Transforms.- 5.1 The Chirp z-Transform Algorithm.- 5.1.1 Real Time Computation of Convolutions and DFTs Using the Chirp z-Transform.- 5.1.2 Recursive Computation of the Chirp z-Transform.- 5.1.3 Factorizations in the Chirp Filter.- 5.2 Rader's Algorithm.- 5.2.1 Composite Algorithms.- 5.2.2 Polynomial Formulation of Rader's Algorithm.- 5.2.3 Short DFT Algorithms.- 5.3 The Prime Factor FFT.- 5.3.1 Multidimensional Mapping of One-Dimensional DFTs.- 5.3.2 The Prime Factor Algorithm.- 5.3.3 The Split Prime Factor Algorithm.- 5.4 The Winograd Fourier Transform Algorithm (WFTA).- 5.4.1 Derivation of the Algorithm.- 5.4.2 Hybrid Algorithms.- 5.4.3 Split Nesting Algorithms.- 5.4.4 Multidimensional DFTs.- 5.4.5 Programming and Quantization Noise Issues.- 5.5 Short DFT Algorithms.- 5.5.1 2-Point DFT.- 5.5.2 3-Point DFT.- 5.5.3 4-Point DFT.- 5.5.4 5-Point DFT.- 5.5.5 7-Point DFT.- 5.5.6 8-Point DFT.- 5.5.7 9-Point DFT.- 5.5.8 16-Point DFT.- 6 Polynomial Transforms.- 6.1 Introduction to Polynomial Transforms.- 6.2 General Definition of Polynomial Transforms.- 6.2.1 Polynomial Transforms with Roots in a Field of Polynomials.- 6.2.2 Polynomial Transforms with Composite Roots.- 6.3 Computation of Polynomial Transforms and Reductions.- 6.4 Two-Dimensional Filtering Using Polynomial Transforms.- 6.4.1 Two-Dimensional Convolutions Evaluated by Polynomial Transforms and Polynomial Product Algorithms.- 6.4.2 Example of a Two-Dimensional Convolution Computed by Polynomial Transforms.- 6.4.3 Nesting Algorithms.- 6.4.4 Comparison with Conventional Convolution Algorithms.- 6.5 Polynomial Transforms Defined in Modified Rings.- 6.6 Complex Convolutions.- 6.7 Multidimensional Polynomial Transforms.- 7 Computation of Discrete Fourier Transforms by Polynomial Transforms.- 7.1 Computation of Multidimensional DFTs by Polynomial Transforms.- 7.1.1 The Reduced DFT Algorithm.- 7.1.2 General Definition of the Algorithm.- 7.1.3 Multidimensional DFTs.- 7.1.4 Nesting and Prime Factor Algorithms.- 7.1.5 DFT Computation Using Polynomial Transforms Defined in Modified Rings of Polynomials.- 7.2 DFTs Evaluated by Multidimensional Correlations and Polynomial Transforms.- 7.2.1 Derivation of the Algorithm.- 7.2.2 Combination of the Two Polynomial Transform Methods.- 7.3 Comparison with the Conventional FFT.- 7.4 Odd DFT Algorithms.- 7.4.1 Reduced DFT Algorithm. N = 4.- 7.4.2 Reduced DFT Algorithm. N = 8.- 7.4.3 Reduced DFT Algorithm. N = 9.- 7.4.4 Reduced DFT Algorithm. N = 16.- 8 Number Theoretic Transforms.- 8.1 Definition of the Number Theoretic Transforms.- 8.1.1 General Properties of NTTs.- 8.2 Mersenne Transforms.- 8.2.1 Definition of Mersenne Transforms.- 8.2.2 Arithmetic Modulo Mersenne Numbers.- 8.2.3 Illustrative Example.- 8.3 Fermat Number Transforms.- 8.3.1 Definition of Fermat Number Transforms.- 8.3.2 Arithmetic Modulo Fermat Numbers.- 8.3.3 Computation of Complex Convolutions by FNTs.- 8.4 Word Length and Transform Length Limitations.- 8.5 Pseudo Transforms.- 8.5.1 Pseudo Mersenne Transforms.- 8.5.2 Pseudo Fermat Number Transforms.- 8.6 Complex NTTs.- 8.7 Comparison with the FFT.- Appendix A Relationship Between DFT and Conyolution Polynomial Transform Algorithms.- A.1 Computation of Multidimensional DFT's by the Inverse Polynomial Transform Algorithm.- A.1.1 The Inverse Polynomial Transform Algorithm.- A.1.2 Complex Polynomial Transform Algorithms.- A.1.3 Round-off Error Analysis.- A.2 Computation of Multidimensional Convolutions by a Combination of the Direct and Inverse Polynomial Transform Methods.- A.2.1 Computation of Convolutions by DFT Polynomial Transform Algorithms.- A.2.2 Convolution Algorithms Based on Polynomial Transforms and Permutations.- A.3 Computation of Multidimensional Discrete Cosine Transforms by Polynomial Transforms.- A.3.1 Computation of Direct Multidimensional DCT's.- A.3.2 Computation of Inverse Multidimensional DCT's.- Appendix B Short Polynomial Product Algorithms.- Problems.- References.

Abstract: A new N = 2n fast Fourier transform algorithm is presented, which has fewer multiplications and additions than radix 2n, n = 1, 2, 3 algorithms, has the same number of multiplications as the Raderi-Brenner algorithm, but much fewer additions, and is numerically better conditioned, and is performed ‘in place’ by a repetitive use of a ‘butterfly’-type structure.

Abstract: The fast Fourier transform algorithm has a long and interesting history that has only recently been appreciated. In this paper, the contributions of many investigators are described and placed in historical perspective.

Abstract: When marine magnetic anomalies can be adequately modeled by two-dimensional magnetic structures within one or more plane layers, many interesting manipulations of both models and anomalies are linear filtering operations [Dean, 1958; Bott, 1967; Black and Scollar, 1969; Schouten, 1971]. Linear filters can be applied quickly and accurately by using the fast Fourier transform algorithm [Cooley and Tukey, 1965]. We have applied this algorithm to two broad classes of physically meaningful filters. The first class consists of the filters that together can synthesize theoretical magnetic anomalies, given a magnetic model. The second class consists of filters that are the inverse of the filters in the first class and can be used to resolve the source of an observed anomaly. We shall describe in some detail the two classes of filters and their application to synthesizing theoretical magnetic anomalies and ultimately to resolving two correlating magnetic anomaly sections of the Eltanin 41 track across an asymmetrically spreading area south of Australia [Weissel and Hayes, 1971, 1972].