Showing papers on "Prime-factor FFT algorithm 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.

A new algorithm for two-dimensional maximum entropy power spectrum estimation

Abstract: A new iterative algorithm for the maximum entropy power spectrum estimation is presented in this paper. The algorithm, which is applicable to two-dimensional signals as well as one-dimensional signals, utilizes the computational efficiency of the fast Fourier transform (FFT) algorithm and has been empirically observed to solve the maximum entropy power spectrum estimation problem. Examples are shown to illustrate the performance of the new algorithm.

A unified treatment of Cooley-Tukey algorithms for the evaluation of the multidimensional DFT

Abstract: In this paper the Cooley-Tukey fast Fourier transform (FFT) algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete Fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or Fourier domain. This general algorithm incorporates both the traditional rectangular row-column and vector-radix algorithms as special cases. This FFT algorithm is shown to result from the factorization of an integer matrix; for each factorization of that matrix, a different algorithm can be developed. This paper presents the general algorithm, discusses its computational efficiency, and relates it to existing multi-dimensional FFT algorithms.

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.

New fast Fourier transform algorithm for linear system analysis applied in molecular beam relaxation spectroscopy

Abstract: The well‐known FFT (Fast Fourier Transform) algorithm of Cooley and Tukey has found widespread application in the field of numerical Fourier transforms and related problems. Nevertheless, the FFT algorithm has to be treated with caution, because in order to avoid aliasing in the frequency domain, it is restricted to the transform of bandlimited functions. A new algorithm has been developed that still has the FFT characteristics, but that is no longer confined to bandlimited functions. The new algorithm has been tested in the case of MBRS (Molecular Beam Relaxation Spectroscopy) linear system analysis.

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.

Floating point roundoff error in the prime factor FFT

Abstract: The prime factor fast Fourier transform (PF FFT), developed by Kolba and Parks, makes use of recent computational complexity results by Winograd to compute the DFT with a fewer number of multiplications than that required by the FFT. Patterson and McClellan have derived an expression for the mean squared error (MSE) in the PF FFT, assuming finite precision fixed point arithmetic. In this paper, we derive an expression for the MSE in the PF FFT, assuming floating point arithmetic. This expression is quite complicated, so an upper bound on the MSE is also derived which is easier to compute. Simulation results are presented comparing the error in the PF FFT with both the derived bound and the error observed in a radix-2 FFT.

An Efficient Two-Dimensional FFT Algorithm

Abstract: A new version of the radix-2 row-column method for computing two-dimensional fast Fourier transforms is proposed. It uses a ``multiple vector'' FFT algorithm to compute the transforms of all the columns in an array simultaneously while avoiding all trivial multiplications. The minicomputer implementation of the algorithm runs faster than the 2 × 2 vector radix FFT algorithm. Analysis of the numbers of complex additions and multiplications required indicate that implementations of the radix-4 row-column FFT and 4 × 4 vector radix FFT on the same minicomputer would run slower than the multiple vector implementation.

Algebraic formulation of the fast Fourier transform

Abstract: A new unified formulation of the fast Fourier transform is presented. It is shown that all FFT algorithms can be derived by different methods of multidimensional array unwrapping. The eight principal decimation in time FFT algorithms are derived. Two algorithms with the desirable properties of sequential input, sequential output and identical computational geometry are also derived. The derivation of the decimation in frequency FFT algorithms is then discussed. While most of the results presented here have been derived earlier using matrix Kronecker products, the present formulation is simpler and more intuitive that the equivalent matrix formulation.

A new prime factor FFT algorithm

Abstract: This paper presents an approach to calculating the discrete Fourier transform (DFT) using a prime factor algorithm (PFA). A very simple indexing scheme is employed that results in a flexible, modular program that very efficiently calculates the DFT in-place. A modification of this indexing scheme gives a new algorithm with the output both in-place and in-order at a slight cost in flexibility. This means only 2N data storage is needed for a length N complex FFT and no unscrambling is necessary. The basic part of a FORTRAN program is given. A speed comparison shows the new algorithm to be faster than both the Cooley-Tukey and the nested Winograd algorithms.

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.

Another discrete Fourier transform computation with small multiplications via the Walsh transform

Abstract: This paper describes another computational algorithm for the discrete Fourier transform(DFT) via the discrete Walsh transform(DWT). The number of multiplications required by this algorithm is approximately NL/9 where N is the number of data points and L is the number of Fourier coefficients desired. This number shows a 33 % decrease against NL/6 in the previous algorithm published by us. The proposed algorithm can be derived by using conventional sampling points in the DFT. The DFT computation via the DWT is superior to the fast Fourier transform(FFT) approach in applications where L is relatively small compared with N and where the Walsh and Fourier coefficients are both desired.

Fixed-point error analysis of Winograd basic DFT algorithms considering correlation between noise sources

Abstract: Recently, Patterson and McClellan [1] have carried out the fixed-point error analysis of winograd Fourier transform algorithm assuming noise sources to be uncorrelated and have derived expressions for output NSR (noise to signal ratio). Such an assumption is not quite valid. A theoretically revised expression without making such an assumption has been obtained in case of Winograd short-length DFT algorithms. This involves the computation of correlation coefficients among noise sources. The revised results are verified by computer simulations and are found to be in close agreement with those predicted by Patterson and McClellan. It is, in general, observed that the basic modules introduce errors higher than those of FFT and require maximum of half a bit to maintain the same degree of precision.

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.

Conditions for the Existence of Fast Number Theoretic Transforms

Abstract: A new theorem that gives necessary and sufficient conditions for the existence of computationally fast number theoretic transforms is presented. The theorem combines the general conditions for the existence of number theoretic transforms in the rings of integers modulo m with two conditions for high computational efficiency.

Algebraic formulation of the fast Fourier transform

Abstract: A new unified formulation of the fast Fourier transform is presented. It is shown that all FFT algorithms can be derived by different methods of multidimensional array unwrapping. The eight principal decimation in time FFT algorithms are derived. Two algorithms with the desirable properties of sequential input, sequential output and identical computational geometry are also derived. The derivation of the decimation in frequency FFT algorithms is then discussed. While most of the results presented here have been derived earlier using matrix Kronecker products, the present formulation is simpler and more intuitive that the equivalent matrix formulation.

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.

On the simulation of residue number systems

Abstract: It is sometimes difficult to simulate on general purpose computers the performance of digital systems that use residue number systems. This paper demonstrates a new technique that makes effective use of a Fast Fourier Transform (FFT) to simulate the basic arithmetic operations required by such number systems. An algorithm for performing such operations which involves the FFT, standard programming techniques and normal arithmetic computations is presented. It is exemplified by consider operations in finite fields. The FFT size is proportional to the exponent of the size of the residue system. Thus there is a logarithmetic relationship between the number system's size and the transform length. Favarable comparisons are given between the FFT mechanization of this approach and one employing a more conventional assembly language implementation.

Evaluation of two-dimensional discrete Fourier transforms via generalized FFT algorithms

Abstract: In this paper two-dimensional fast Fourier transforms (FFT's) are expressed as special cases of a generalization of the one-dimensional Cooley-Tukey algorithm. This generalized algorithm allows the efficient evaluation of discrete Fourier transforms (DFT's) of rectangularly sampled sequences, hexagonally sampled sequences and arbitrary periodically sampled sequences. Significant computational savings can be realized using this generalized algorithm when the periodicity matrix of the sequence is highly composite. Alternate factorizations of the periodicity matrix lead to different FFT algorithms, including the row-column decomposition and the vector-radix algorithm. This paper will present a generalized DFT, derive the general 2-D Cooley-Tukey algorithm and conclude by interpreting several 2-D FFT algorithms in terms of the generalized one.

A new algorithm for one-dimensional and two-dimensional maximum entropy power spectrum estimation

Abstract: A new iterative algorithm for the maximum entropy power spectrum estimation is presented in this paper. The algorithm which is applicable to two-dimensional signals as well as one-dimensional signals, utilizes the computational efficiency of the Fast Fourier Transform (FFT) algorithm and has been empirically observed to solve the maximum entropy power spectrum estimation problem. Examples are shown to illustrate the performance of the new algorithm.