Showing papers on "Prime-factor FFT algorithm published in 1989"
Book•
01 Dec 1989
TL;DR: In this article, the authors introduce multiplicative Fourier transform algorithms (MFTA) for abstract algebra and discuss the prime case and the Product of Two Distinct Primes (P2) case.
Abstract: Contents: Introduction to Abstract Algebra.- Tensor Product and Stride Permutation.- Cooley-Tukey FFF Algorithms.- Variants of FFT Algorithms and Their Implementations.- Good-Thomas PFA.- Linear and Cyclic Convolutions.- Agarwal-Cooley Convolution Algorithm.- Introduction to Multiplicative Fourier Transform Algorithms (MFTA).- MFTA: The Prime Case.- MFTA: Product of Two Distinct Primes.- MFTA: Transform Size N = Mr. M-Composite Integer and r-Prime.- MFTA: Transform Size N = p2.- Periodization and Decimation.- Multiplicative Character and the FFT.- Rationality.- Index.
230 citations
TL;DR: In this article, the split-radix algorithm for the discrete Fourier transform (DFT) of length 2/sup m/ is considered, and it is shown that whenever a radix-p/p/sup 2/ outperforms a single-Radix algorithm, then a Radix-P/P/Sup 2/2/ algorithm will outperform both of them.
Abstract: The split-radix algorithm for the discrete Fourier transform (DFT) of length-2/sup m/ is considered. First, the reason why the split-radix algorithm is better than any single-radix algorithm on length-2/sup m/ DFTs is given. Then, the split-radix approach is generalized to length-p/sup m/ DFTs. It is shown that whenever a radix-p/sup 2/ outperforms a radix-p algorithm, then a radix-p/p/sup 2/ algorithm will outperform both of them. As an example, a radix-3/9 algorithm is developed for length-3/sup m/ DFTs. >
57 citations
TL;DR: A description is given of a novel algorithm, the fast Fourier transform in part (FFTP), for the computation of the discrete pseudo-Wigner distribution (DPWD), which reduces the computational cost by making full use of symmetries and removing redundancies in the FFTP computation.
Abstract: A description is given of a novel algorithm, the fast Fourier transform in part (FFTP), for the computation of the discrete pseudo-Wigner distribution (DPWD). The FFTP computes the cosine and sine parts of the discrete Fourier transform (DFT) separately by employing real inverse sinusoidal twiddle factors. Unlike the conventional methods which directly utilize the complex DFT, the FFTP yields real output since the DPWD is always real. In addition, the new method reduces the computational cost by making full use of symmetries and removing redundancies in the FFTP computation. The authors also describe a simple algorithm for computing the discrete Hilbert transform (DHT) to produce the nonaliased DPWD. A pipeline structure for real-time and a bulk processing technique for offline implementations of the method are presented. >
43 citations
23 May 1989
TL;DR: A psychophysically justified bit allocation algorithm for use with subband image coding systems and is superior to the minimum-mean-square-error algorithm at low bit rates.
Abstract: The authors present a psychophysically justified bit allocation algorithm for use with subband image coding systems. This algorithm is also appropriate for DCT and DFT (discrete cosine and discrete Fourier transform)-based analysis-synthesis systems, as well as many others. The subjective performance of the algorithm was compared with the performance of a related algorithm that seeks to minimize the mean-square coding error. A two-dimensional analysis-synthesis system of J.P. Princen and A.B. Bradley (IEEE Trans. on Acoust. Speech, and Signal Proc., vol.ASSP-34, no.5, p.1153-1161, Oct. 1986) was used for the comparison. It is found that the psychophysically justified algorithm is superior to the minimum-mean-square-error algorithm at low bit rates. >
40 citations
TL;DR: In the letter a fast and efficient algorithm is presented for calculating both the DFT and the WHT through the factorisation of the intermediate transform into a product of sparse matrices.
Abstract: In the letter a fast and efficient algorithm is presented for calculating both the DFT and the WHT. This is achieved through the factorisation of the intermediate transform into a product of sparse matrices. The algorithm can implemented using a single butterfly structure, and is amenable for both software and hardware implementations.
40 citations
TL;DR: A new algorithm for the fast computation of the discrete Fourier transform (DFT) is introduced, called the conjugate pair FFT (CPFFT), which is used to compute a length-2m DFT.
Abstract: A new algorithm for the fast computation of the discrete Fourier transform is introduced. The algorithm, called the conjugate pair FFT (CPFFT), is used to compute a length-2m DFT. The number of multiplications and additions required by the CPFFT is less than that required by the SRFFT algorithm.
31 citations
TL;DR: A structure theorem is devised to construct systematically various vector-radix decimation-in-frequency FFT algorithms from their 1-D counterparts.
Abstract: A general form of the matrix representation for multidimensional, vector-radix, fast Fourier transform (FFT) algorithms using decimation-in-frequency is presented. A structure theorem is devised to construct systematically various vector-radix decimation-in-frequency FFT algorithms from their 1-D counterparts. Logic diagrams are provided to facilitate the software and hardware implementation of the algorithms. The computational complexity of several of the algorithms is considered. >
28 citations
27 Mar 1989
TL;DR: This paper presents a recursive, radix two by two, fast algorithm for computing the two dimensional discrete cosine transform (2D-DCT), which allows the generation of the next higher order 2D- DCT from four identical lower order 2Ds with the structure being similar to the twodimensional fast Fourier transform.
Abstract: This paper presents a recursive, radix two by two, fast algorithm for computing the two dimensional discrete cosine transform (2D-DCT). The algorithm allows the generation of the next higher order 2D-DCT from four identical lower order 2D-DCT's with the structure being similar to the two dimensional fast Fourier transform (2D-FFT). As a result, the method for implementing this recursive 2D-DCT requires fewer multipliers and adders than other 2D-DCT algorithms.
23 citations
Journal Article•
22 citations
TL;DR: A new fast algorithm for computing the two-dimensional discrete Hartley transform that requires the lowest number of multiplications compared with other related algorithms is presented.
Abstract: A new fast algorithm for computing the two-dimensional discrete Hartley transform is presented. This algorithm requires the lowest number of multiplications compared with other related algorithms.
18 citations
TL;DR: The author states that generally, the vector split-radix method provides a significant reduction in the number of complex multiplications required to implement a two-dimensional discrete Fourier transform.
Abstract: The complete equations are presented for the first stage of the two-dimensional vector split-radix decimation-in-frequency fast Fourier transform algorithm using a structural approach. The computational complexity of the algorithm is discussed and compared to other published results. The author states that generally, the vector split-radix method provides a significant reduction in the number of complex multiplications required to implement a two-dimensional discrete Fourier transform. >
TL;DR: A fourth theorem is given, indicating an alternate order of generating the index pairs for swapping, and a second permutation algorithm results, which uses the same principles as the first and differs principally in that in its innermost loop, one of the pair of indexes is usually generated by an integer increment.
Abstract: Based on three previously published theorems and an algorithm for the digit-reversal permutation required by fast transform algorithms, a fourth theorem is given, indicating an alternate order of generating the index pairs for swapping, and a second permutation algorithm results. This algorithm uses the same principles as the first and differs principally in that in its innermost loop, one of the pair of indexes is usually generated by an integer increment (i:=i+n). This will result in slightly faster execution on most computers. >
01 May 1989
TL;DR: This paper shows how a rectangular array of N CORDIC (co-ordinate digital computer) processing elements can be used to carry out an efficient two-dimensional systolic implementation of the N-point DFT, offering highly attractive throughput rates in relation to other N-processor solutions.
Abstract: A number of systolic architectures have appeared over the past few years for performing the discrete Fourier transform (DFT) and fast Fourier transform (FFT) algorithms, using both linear and orthogonal processing networks. The paper shows how a rectangular array of N CORDIC (co-ordinate digital computer) processing elements can be used to carry out an efficient two-dimensional systolic implementation of the N-point DFT, offering highly attractive throughput rates in relation to other N-processor solutions, such as the conventional linear systolic array. >
IBM1
TL;DR: In this article, a multi-dimensional discrete Fourier series method has been developed to calculate the coupling capacitances in VLSI circuits, which is related to the known electric potential distributions by Laplace equation.
Abstract: A multi-dimensional discrete Fourier series method has been developed to calculate the coupling capacitances in VLSI circuits. In the present method a scalar potential Φ( r ) is generated, in discrete Fourier series form, which is related to the known electric potential distributions by Laplace equation. By using the multiple radix Fast Fourier Transform (FFT) algorithm to reduce the CPU time, the electric field strength can be obtained from the gradient of the scalar potential. The calculated 2-D and 3-D results generally compare well with the conventional approach results. A program has been developed and numerical results are discussed.
TL;DR: Critical algorithm design issues are discussed, necessary machine performance measurements are identified and made, and the performance of the developed FFT programs are measured.
Abstract: The Fast Fourier Transform is a mainstay of certain numerical techniques for solving fluid dynamics problems. The Connection Machine CM-2 is the target for an investigation into the design of multidimensional SIMD parallel FFT algorithms for high performance. Critical algorithm design issues are discussed, necessary machine performance measurements are identified and made, and the performance of the developed FFT programs are measured. Our FFT programs are compared to the currently best Cray-2 FFT library program, CFFT2.
08 May 1989
TL;DR: In this paper, a new interpolation scheme that allows the frequency and amplitude of a sinusoid to be estimated with high accuracy is proposed. But the method is based on the fact that spectral convolution introduces spectral leakage that can be observed at the output of an FFT.
Abstract: The problem with using the FFT to do spectral estimation is that only a sampled version of the discrete Fourier transform of the input signal is provided To do parameter estimation on a signal with frequency that does not appear at the output of the FFT, one has to perform spectral interpolation to approximate the original continuous spectrum The author introduces a new interpolation scheme that allows the frequency and amplitude of a sinusoid to be estimated with high accuracy The method is based on the fact that spectral convolution introduces spectral leakage that can be observed at the output of an FFT With the minimum four-sample Blackman-Harris window, this method yields better results than the conventional parabola-fitting technique The largest error occurs when the frequency is halfway between two frequency bins Higher-order Taylor series could be used to improve the estimation in the range of 045 >
TL;DR: The authors investigated the implementation aspects of the fast Hartley transform (FHT) in both software and hardware and describe the modifications required to convert existing fast Fourier transform programs to execute FHTs, showing the ease with which these modifications can be implemented.
Abstract: The authors investigated the implementation aspects of the fast Hartley transform (FHT) in both software and hardware. They describe the modifications required to convert existing fast Fourier transform (FFT) programs to execute FHTs, showing the ease with which these modifications can be implemented. They compare execution time and memory storage requirements of both transforms and present power spectrum calculation and convolution as illustrative examples to compare the performances of the two transform techniques. They also give a comparative survey of the performances of various microprocessors and digital signal processors in FFT and FHT computation. >
TL;DR: In this article, the authors derived a duality equation relating the extension functions introduced in the extended function FFT (EF-FFT) method to conventional window functions and showed that signals with high-frequency content only within the observation window are best analyzed with EF-FFT methods and signals with time-distributed spectral components (e.g., speech) are best analysed with conventional FFT methods.
Abstract: The periodicity assumption implicit in fast Fourier transform (FFT) techniques can be utilized through time-domain prealiasing to obtain the spectral components of infinite-duration time-domain reflectometry signals when they can be modeled, outside the observation window, with step and/or exponential functions. The technique is shown to be more accurate than both conventional windowing and the other FFT approaches described in the literature for analysis of steplike signals. The duality equation relating the extension functions introduced in the extended function FFT (EF-FFT) method to conventional window functions is derived. Using this relation, it is shown that signals with high-frequency content only within the observation window are best analyzed with EF-FFT methods and that signals with time-distributed spectral components (e.g., speech) are best analyzed with conventional FFT methods. >
TL;DR: In this article, it is shown that the fast Fourier transform (FFT) combines naturally with Simpson's rule for Sommerfeld-type integral computation, and several examples are provided to illustrate the process.
Abstract: It is shown that the fast Fourier transform (FFT) combines naturally with Simpson's rule for Sommerfeld-type integral computation. The principal advantage of using the FFT is that a single subroutine call yields a set of sample values of an integral (i.e. the integral for various values of an integrand parameter). Such samples could be useful in themselves. In other applications Sommerfeld integrals represent Green's functions nested within other spatial integrals, so samples from the FFT might be useful in approximating the outernested integral. Several examples are provided to illustrate the process. >
21 Aug 1989
TL;DR: This paper reports an explanation of an intricate algorithm in the terms of a potentially mechanisable rigorous-development method, using notations and techniques of Sheeran and Bird and Meertens and claiming that these techniques are applicable to digital signal processing circuits.
Abstract: This paper reports an explanation of an intricate algorithm in the terms of a potentially mechanisable rigorous-development method. It uses notations and techniques of Sheeran [1] and Bird and Meertens [2, 3]. We have claimed that these techniques are applicable to digital signal processing circuits, and have previously applied them to regular array circuits [4, 5, 6].
TL;DR: Two multiplicative FFT algorithms have operational counts close to the Winograd algorithm, but they have a better structure, which simplifies their implementation and are suited for conventional serial machines.
08 May 1989
TL;DR: The two-factor Cooley-Tukey FRFT algorithm is developed and expressed in terms of matrix factorization using Kronecker products, which is generalized to any number of factors.
Abstract: In many applications, it is desirable to have a fast algorithm (FRFT) for the computation of the real discrete Fourier transform (RDFT) for any number of data points. To achieve this, the two-factor Cooley-Tukey FRFT algorithm is developed and expressed in terms of matrix factorization using Kronecker products. This is generalized to any number of factors. Each factor M involves the computation of size M RDFTs, which is carried out by the best size M FRFT algorithm available. >
TL;DR: An efficient algorithm for computing the discrete cosine transform (DCT) is presented, based on an index mapping which converts an odd-length DCT to a realvalued DFT of the same length using permutations and sign changes only.
Abstract: In this letter, an efficient algorithm for computing the discrete cosine transform (DCT) is presented. It is based on an index mapping which converts an odd-length DCT to a realvalued DFT of the same length using permutations and sign changes only. The real-valued DFT can then be computed by efficient real-valued FFT algorithms such as the prime factor algorithm. The algorithm is more efficient than an earlier one because no postmultiplications are required.
23 May 1989
TL;DR: The authors describe the implementation of real and complex FFT (fast Fourier transform) algorithms on the Motorola DSP96002, a general-purpose, dual-bus IEEE standard floating-point digital signal processor that provides the basis for efficient implementation of FFTs and other fast transforms.
Abstract: The authors describe the implementation of real and complex FFT (fast Fourier transform) algorithms on the Motorola DSP96002. The DSP96002 is a general-purpose, dual-bus IEEE standard floating-point digital signal processor (DSP). At a 74-ns instruction cycle, the DSP96002 implements a 1024-point real FFT in 0.905 ms and a 1024-point complex FFT in 1.55 ms. This performance is achieved by calculating up to three floating-point results in a single instruction cycle, or 40.5 MFLOPS peak. A radix-2 FFT butterfly is executed every four cycles, an average of 33.75 IEEE MFLOPS. The instruction set and architecture of the DSP96002 provide the basis for efficient implementation of FFTs and other fast transforms, such as the discrete Walsh-Hadamard transform, discrete cosine transform, and discrete Hartley transform. >
01 Jan 1989
TL;DR: The main idea is to use the additive structure of the indexing set Z/N to define mappings of the input and output data vectors into 2-dimensional arrays which, when combined with these mappings, compute the N-point FFT.
Abstract: In the following two chapters, we will concentrate on algorithms for computing FFT of size a composite number N. The main idea is to use the additive structure of the indexing set Z/N to define mappings of the input and output data vectors into 2-dimensional arrays. Algorithms are then designed, transforming 2-dimensional arrays which, when combined with these mappings, compute the N-point FFT. The stride permutations of chapter 2 play a major role.
01 Aug 1989
TL;DR: In this paper, a new least squares algorithm is proposed and investigated for fast frequency and phase acquisition of sinusoids in the presence of noise, which is a special case of more general, adaptive parameter estimation techniques.
Abstract: A new least squares algorithm is proposed and investigated for fast frequency and phase acquisition of sinusoids in the presence of noise. This algorithm is a special case of more general, adaptive parameter-estimation techniques. The advantages of the algorithms are their conceptual simplicity, flexibility and applicability to general situations. For example, the frequency to be acquired can be time varying, and the noise can be non-Gaussian, nonstationary and colored. As the proposed algorithm can be made recursive in the number of observations, it is not necessary to have a priori knowledge of the received signal-to-noise ratio or to specify the measurement time. This would be required for batch processing techniques, such as the fast Fourier transform (FFT). The proposed algorithm improves the frequency estimate on a recursive basis as more and more observations are obtained. When the algorithm is applied in real time, it has the extra advantage that the observations need not be stored. The algorithm also yields a real time confidence measure as to the accuracy of the estimator.
TL;DR: A new fast algorithm for computing the two-dimensional discrete Fourier transform DFT(2n; 2) using the fast discrete cosine transform algorithm that uses only real multiplications, which is more suitable for real input data.
Abstract: We present a new fast algorithm for computing the two-dimensional discrete Fourier transform DFT(2n; 2) using the fast discrete cosine transform algorithm. The algorithm has a lower number of multiplications and additions compared with other published algorithms for computing the two-dimensional DFT. Because it uses only real multiplications, the algorithm is more suitable for real input data.
TL;DR: On large scientific computers where the multiplications can be overlapped with the additions, the prime factor algorithm without nesting remains the fastest and is the easiest to implement.
TL;DR: It is found that the WFTA implementation does save CPU time when implemented on a general-purpose computer for the following reasons: the number of multiplications is far fewer than for common algorithms.
Abstract: The author introduces a two-dimensional Winograd Fourier transform algorithm (WFTA) technique and compares it with more traditional fast Fourier transform (FFT) implementations. Techniques for programming the WFTA in two dimensions are introduced. For completeness, enumerations of the Winograd small-n transposes, required when applying these techniques, are included. It is found that the WFTA implementation does save CPU time when implemented on a general-purpose computer for the following reasons: (1) the number of multiplications is far fewer than for common algorithms; (2) integer arithmetic can be used, at least for several stages of input additions, with no loss of accuracy; and (3) the fact that many operations are written out explicitly allows a programmer to save on the computation of indexes. >
01 Jan 1989
TL;DR: This chapter reviews fast Fouriertransform methods, a collective term for a number of efficient algorithms developed to compute the discrete Fourier transform (DFT) and the inverse discrete Fouriers transform (IDFT).
Abstract: This chapter reviews fast Fourier transform methods. The fast Fourier transform (FFT) is a collective term for a number of efficient algorithms developed to compute the discrete Fourier transform (DFT) and the inverse discrete Fourier transform (IDFT). It is not a transform in its own right, and the understanding and interpretation of FFT methods depend fundamentally on the theory and properties of the DFT. It is not difficult to run DFT programs on a microcomputer, and FFT algorithms can be quite satisfactorily demonstrated without excessive demands on memory or processing rates. Although a sound understanding of the principles and properties of the DFT is necessary to apply the FFT and interpret the output of FFT programs, in many applications, an FFT subroutine can be used as a black box without a detailed knowledge of the algorithm employed. FFT algorithms can be used to great advantage in a wide range of signal-processing applications. These include Fourier analysis and synthesis of signals and spectrum analysis in general. FFT methods can be used to implement FIR digital filters.