Showing papers on "Cyclotomic fast Fourier transform published in 1981"

Fast Fourier transform and convolution algorithms

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.

An in-place, in-order prime factor FFT algorithm

Abstract: This paper presents a Fortran program that calculates the discrete Fourier transform using a prime factor algorithm. A very simple indexing scheme is employed that results in a flexible, modular algorithm that efficiently calculates the DFT in-place. A modification of this algorithm gives the output both in-place and in-order at a slight cost in flexibility. A comparison shows it to be faster than both the Cooley-Tukey algorithm and the Winograd nested algorithm.

Dual algorithms for fast calculation of the Fourier-Bessel transform

Abstract: This paper presents a new procedure for the fast calculation of the Fourier-Bessel transform. The computation is performed in a "dual" mode and involves two matched algorithms, the first of which provides the lower order components while the second yields intermediate and higher order samples of the transform. The first algorithm is based on the Fourier-selection-summation (FSS) method derived in a previous study, while the second relies on the Bessel function large argument asymptotic expansion. It is shown that the dual procedure constructed along these lines requires a computation time which is about four times that needed by a single N point fast Fourier transform.

Area-Time Optimal VLSI Networks for Computing Integer Multiplications and Discrete Fourier Transform

Abstract: In this paper we present a class of VLSI networks for computing the Discrete Fourier Transform and the product of two N-bit integers. These networks match, within a constant factor, the known theoretical lower-bound O(N2) to the area × (time)2 measure of complexity. While this paper's contribution is mainly theoretical, it points toward very practical directions: we show how to design multipliers with area A = O(N) and time T=O(√N) on one hand, and A=0((N/log2N)2), T = O(log2N) on the other. Both of these designs should be contrasted with the currently available multipliers, whose performances are A=O(N), T=O(N) or even A=O(N2), T=O(N).

New polynomial transform algorithms for multidimensional DFT's and convolutions

Abstract: This paper introduces a new method for the computation of multidimensional DFT's by polynomial transforms. The method, which maps mtiltidimensional DFT's into one-dimensional odd-time DFT's by use of inverse polynomial transforms, is shown to be significantly more efficient than the conventional row-column method from the standpoint of the number of arithmetic operations and quantization noise. The relationship between DFT and convolution algorithms using polynomial transforms is clarified and new convolution algorithms with reduced computational complexities are proposed.

A vector implementation of the Fast Fourier Transform algorithm

Abstract: A recent article in this journal by D. G. Korn and J. J. Lambiotte, Jr. discusses implementations of the FFT algorithm on the CDC STAR-100 vector computer. The 'Pease'-algorithm is recommended in cases when only a few transforms can be performed simultaneously. We show how the use of a different algorithm and of trigonometric tables will lead to more than three times faster execution times. The times for large transforms increase only about 39% if the tables are eliminated in order to save storage.

The fast calculation of the exact distribution of pearson's chi-squared and of the number of repeats within the cells of a mltltinomial by using a fast fourier transform

Abstract: (1981). The fast calculation of the exact distribution of pearson's chi-squared and of the number of repeats within the cells of a mltltinomial by using a fast fourier transform. Journal of Statistical Computation and Simulation: Vol. 14, No. 1, pp. 71-78.

Fast computation of Fourier transforms at arbitrary frequencies

Abstract: An algorithm is proposed for computing the Fourier Transform (FT) of a uniformly sampled signal at arbitrary frequencies. In most of the applications, the algorithm retains the computational efficiency of the Fast Fourier Transform (FFT) algorithm. The method is based on the fact that the FT at an arbitrary frequency can be expressed as a weighted sum of its Discrete Fourier Transform (DFT) coefficients. In the proposed method, these weights are suitably approximated so that the desired FT is very nearly the sum of (i) a few dominant terms of the sum of the DFT which are computed directly, and (ii) the DFT of a new sequence obtained by multiplying the original sequence with a sawtooth function. The number of directly computed terms is so chosen that the error of approximation does not exceed the specified limits. The computational aspects of the algorithm and its error behavior with typical signals have been critically examined.

Winograd's Discrete Fourier Transform Algorithm

Abstract: The new DFT algorithm of S. Winograd is developed and presented in detail. This is an algorithm which uses about 1/5 of the number of multiplications used by the Cooley-Tukey algorithm and is applicable to any order which is a product of relatively prime factors from the following list : 2, 3, 4, 5, 7, 8, 9, 16. The algorithm is presented in terms of a series of tableaus one for each term in this list which are convenient, compact, graphical representations of the sequence of arithmetic operations in the corresponding parts of the algorithm. Using these in conjunction with Tables 4.5, 6, makes it relatively easy to apply the algorithm and evaluate its performance. The development of the subject is organized in a way which allows extensive skipping on a first reading.

Architecture of the fast Walsh-Hadamard and fast Fourier transforms with charge transfer devices

Abstract: This paper presents a novel approach for charge transfer device (CTD) implementation of the fast Walsh-Hadamard and fast Fourier transforms. This is achieved by first expressing the Hadamard matrix as a power of a matrix, which allows for efficient CTD implementation. These results are then used in CTD implementation of fast Fourier transforms. The errors which accrue on using CTDs for implementing the Walsh-Hadamard transform are discussed in terms of a dispersion estimate.

m-adic (polyadic) invariance and its applications

Abstract: The concept of m-adic invariance allows approximation of a linear time-invariant (LTI) system by a linear m-adic invariant (LMI) system or, equivalently, approximation of a circulant matrix by a super-circulant matrix. The approximation reduces the number of multiplies required for computing N-point cyclic convolution to 2(\log_{m}N - 1)N , where N = mn. The error introduced by the approximation can be removed, if desired, by subsequent processing. In one concrete case, determination of a small number of noncontiguous frequencies, this approach-approximation and subsequent correction-can effect substantial savings in a number of multiplies compared to both fast Fourier transform (FFT) algorithm and direct discrete Fourier transform (DDFT). These applications are preceded by a tutorial presentation of concepts which are basic to m-adic invariant systems.

Fast polynomial transform and its implementation by computer

Abstract: A fast polynomial transform (FPT) algorithm for computing two-dimensional cyclic convolutions on a general-purpose computer is demonstrated and compared with the FFT approach. An FPT program for two-dimensional convolutions written in FORTRAN is shown to be 20% faster than the conventional FFT algorithm. This higher speed advantage makes the FPT algorithm a candidate for many two-dimensional digital image filtering applications.

Separation of kernel in the hexagonal discrete fourier transform

Abstract: For the two-dimensional signal with circular band-limiting in frequency space, a square arrangement of sampling points in the real space is usually used. Assuming, however, that the signal is band-limited in a hexagonal region, being regarded as periodic and adopting the triangular sampling point arrangmeent in real space, the number of sampling points can be reduced by 13.4% that in the usual method. Mersereau has derived a discrete Fourier transform (DFT) for a two-dimensional signal hexagonal band-limited both in real and frequency spaces. In his method, however, separation of the Fourier kernel is impossible and Rivard's FFT algorithm is not applicable to the computation of hexagonal DFT. The authors introduce a periodic extension vector system and sampling point generating vector system. By a generalized method, the two-dimensional DFT is reformulated. It is shown that the kernel can be given a separable expression by suitable choice of coordinates as in the square DFT, actually presenting the method of determining the coordinate. When the kernel of a hexagonal DFT is separable, the computation reduces to that of the one-dimensional DFT. This permits the application of already developed FFT algorithms, enlarging the range of utilization of the hexagonal DFT.

Fourier transform algorithm implementations on a general-purpose microprocessor

Abstract: The Fourier representation of sequences plays a key roll in the analysis, the design, and the implementation of digital signal processing algorithms. The existence of very efficient algorithms for computing the Fourier transforms have expanded the importance of Fourier analysis in digital signal processing. To indicate the importance of efficient computational schemes, evaluation of two well-known algorithms - the Cooley-Tukey fast Fourier transform and complex general-N Winograd Fourier transform - were implemented on a general-purpose, high-speed, digital microprocessor - the MC68000. The Despain very fast Fourier algorithm was studied as well. Complexity measures for Fourier transforms, or the relative executional time of an implemented algorithm, have generally been based on the number of multiplications and additions required. For this reason, algorithmic improvements have primarily consisted of reduction in the number of multiplications and additions. However, large amounts of accessing and storing of data, as well as loop control overhead, are inherent in the implementation of these algorithms. Comparisons of the three algorithms as well as numerical versus data transfer operations are presented for a specific microprocessor implementation.

Inverse polynomial transform algorithms for DFTs and convolutions

Abstract: In this paper, we introduce a new fast computation algorithm for multidimensional DFTs. This method uses inverse polynomial transforms to perform an efficient mapping of multidimensional DFTS into one-dimensional DFTs in a way similar to earlier polynomial transform techniques, but with all operations performed in reversed order. This is shown to yield fast DFT algorithms which retain the basic advantages related to the use of polynomial transforms while allowing a significant reduction in round-off noise. We then combine the direct and inverse polynomial transform methods to derive new fast algorithms for multidimensional convolutions.

Addendum to "A New Hybrid Algorithm for Computing a Fast Discrete Fourier Transform"

Abstract: Recently,1 the authors proposed a hybrid algorithm for computing the discrete Fourier transform (DFT) of certain long transform lengths. In that technique, a Winograd-type algorithm was used in conjunction with the Mersenne prime-number theoretic transform to perform a DFT. Even though this technique requires fewer multiplications than either the standard fast Fourier transform (FFT) or Winograd's more conventional algorithm, it increases the number of additions considerably. In this letter it is proposed to use Winograd's algorithm for computing the Mersenne prime-number theoretic transform in the transform portion of the hybrid algorithm. It is shown that this can reduce significantly the number of additions while still maintaining about the same number of multiplications.

Hierarchical discrete systems and realisation of parallel algorithms

Abstract: The subject of the paper is discrete systems (DS) represented by a set of elements and a set of operations. In particular these operations are permutations of DS elements. The DS is represented by subsystems with sufficient small number of elements and operations of the same kind. The subsystems are distributed at hierarchical levels. The subsystems of each level can execute their operations in parallel. On the basis of this approach algorithms are constructed for parallel realisation of the permutations in DS using a generating set of permutations. A theoretical group interpretation of the well-known fast Fourier transform is made, which shows that this approach can be applied to a wider class of transformations.

Comments on "Very Fast Fourier Transform Algorithms Hardware for Implementation"

Abstract: The object of this correspondence is to point out and correct some errors which have occurred in the above paper.1

Corner-turn complexity properties of polynomial transform 2D convolution methods

Abstract: This paper examines the matrix data re-order requirements of a variety of polynomial transform 2D convolution methods which can be employed to efficiently accommodate large field problems. The results indicate that several power-of-2 length polynomial transform methods developed by Nussbaumer allow one to totally avoid the row-column data corner-turn commonly encountered in Fourier Transform 2D convolution methods, while also providing significantly reduced computational complexity. Execution time comparison with an FFT reference base is made assuming the use of general register and array processor units and use of recently developed matrix-transpose methods by Eklundh and Ari to support 2D Fourier Transform corner-turn requirements. These results demonstrate a 2-4 times throughput performance improvement for use of the polynomial transform method in place of the 2D Fourier Transform approach to circularly convolve large 2D fields in the range 1024×1024 to 8192×8192.