Showing papers on "Split-radix FFT algorithm published in 1981"

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 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 new radix-6 FFT algorithm

Abstract: Using complex numbers of the form a + b μ (where μ is a complex cube root of unity), a radix-6 FFT algorithm in which the component six-point DFT's do not require any multiplication is developed. This number system was used by Dubois and Venetsanopoulos to implement radix-3 FFT. The number of arithmetic operations for the new algorithm is compared with those of standard radix-6, radix-2, and radix-4 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.

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.

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.

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.