Showing papers on "Cyclotomic fast Fourier transform published in 1989"
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
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 fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT), where the computation complexity can be greatly reduced from two complex FFTs into two real FHTs.
Abstract: A fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT). Instead of the conventional fast Fourier transform (FFT) approach, the processing is carried out entirely in the real domain. Also, since many efficient FHT algorithms exist, the computation complexity can be greatly reduced from two complex FFTs into two real FHTs. >
34 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
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
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. >
18 citations
TL;DR: In this paper, the switched-capacitor realization of the discrete Fourier transform (DFT) is treated as well as the inverse DFT (1DFT), and the output of the DFT has a sinusoidal waveform including the amplitude and phase information of the required spectra.
Abstract: The switched-capacitor realization of the discrete Fourier transform (DFT) is treated in this paper as well as the inverse discrete Fourier transform (1DFT). The output of the DFT has a sinusoidal waveform including the amplitude and phase information of the required spectra. These spectra are given simultaneously and almost in real time. The output of the 1DFT is given merely by adding DFT outputs. Furthermore, the circuit configuration of this system-from input to DFT, from DFT to 1DFT, and from 1DFT to output-is a very simple configuration constructed by a non-recursive filter circuit.
16 citations
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. >
16 citations
TL;DR: In this article, an algorithm for the determination of the transfer function of a singular state-space system is proposed, which utilizes the discrete Fourier transform (DFT) and has low computational complexity.
Abstract: An algorithm for the determination of the transfer function of a singular state-space system is proposed. This algorithm utilizes the discrete Fourier transform (DFT) and has low computational complexity. The simplicity and efficiency of the method are illustrated by two examples. >
15 citations
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: An algorithm is presented for factoring Fourier matrices into products of bidiagonal matrices, which make possible discrete Fourier transform (DFT) computation via a sequence of local, regular computations.
Abstract: An algorithm is presented for factoring Fourier matrices into products of bidiagonal matrices. These factorizations have the same structure for every n and make possible discrete Fourier transform (DFT) computation via a sequence of local, regular computations. A parallel pipeline technique for computing sequences of k-point DFTs, for every k >
TL;DR: The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers.
Abstract: Some new results for finding all convenient moduli m for a complex numbertheoretic transform with given transform length n and given primitive nth root of unity modulo m are presented. The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers.
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].
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 algorithm is introduced for computing the multidimensional finite Fourier transform and offers a substantial reduction in the computational complexity.
Abstract: An algorithm is introduced for computing the multidimensional finite Fourier transform. The algorithm can be applied to data samples of any size. In most cases, it offers a substantial reduction in the computational complexity. >
01 Jan 1989
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: 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. >
23 May 1989
TL;DR: A method is presented for computing the discrete Fourier transform (DFT) of data compressed using vector quantization (VQ), suitable for both one-dimensional and multidimensional DFTs or, in general, any linear process.
Abstract: A method is presented for computing the discrete Fourier transform (DFT) of data compressed using vector quantization (VQ). The VQ compressed data are not reconstructed before use; instead, a codebook that has been processed with the DFT (discrete Fourier transform) algorithm is used for VQ reconstruction. An overlap-and-add technique is used to combine the processed reconstruction codebook vectors to give the DFT directly. The technique is suitable for both one-dimensional and multidimensional DFTs or, in general, any linear process. The technique is called the computation compression technique (CCT). The CCT implementation yields exactly the same result as if the compressed data had been reconstructed and the DFT performed on the data directly. CCT convolution on a 68020/6881-based computer is described. Speedups of two orders of magnitude are obtained. >
06 Dec 1989
TL;DR: Algorithms to compute a N -dimensional Discrete Fourier Transform in two and N -dimensions are presented and the explicit congruences for minimal number of lines covering a mul-tidimensional grid are given.
Abstract: This paper presents algorithms to com-pute a N -dimensional Discrete Fourier Transform. First existing algorithms are surveyed. Multiplicative symmetrized and orbit exchange algorithm is presented. Then line algorithm to compute Multidi-mensional DFT is derived in two and N -dimensions. The explicit congruences for minimal number of lines covering a mul-tidimensional grid is given, when the data size in one dimension is equal to the power of a prime number.
TL;DR: In this article, the nature of index transforms is explored using group-theoretical ideas, where the computation of m-dimensional "long" DFT's or convolutions can be transfered to the parallel computation of n-dimensional'short' DFTs or convolution (n>m).
Abstract: Index transforms ofm-dimensional arrays inton-dimensional arrays play a significant role in many fast algorithms of multivariate discrete Fourier transforms (DFT's) and cyclic convolutions. The computation ofm-dimensional "long" DFT's or convolutions can be transfered to the parallel computation ofn-dimensional "short" DFT's or convolutions (n>m). In this paper, the nature of index transforms is explored using group-theoretical ideas. We solve the open problems concerning index transforms posed recently by Hekrdla [5, 6].
23 May 1989
TL;DR: A parallel adder configuration that is much faster than the usual serial adder is proposed and a scheme for fast reordering of the input data that increases the reordering speed without increasing the memory size is also proposed.
Abstract: A fast implementation of recursive DFTs (discrete Fourier transforms) is presented. It only needs (N-1)/2 real multiplications to compute all N frequency components. A factor R/sub T/ is introduced. If the ratio T/sub m//T/sub a/ of the multiplier and adder periods is greater than R/sub T/, this scheme is faster than the FFT (fast Fourier transform). The error and signal-to-noise ratio are studied. A parallel adder configuration that is much faster than the usual serial adder is proposed. A scheme for fast reordering of the input data that increases the reordering speed without increasing the memory size is also proposed. >
26 Jun 1989
TL;DR: A spectrum analyzer is presented for three-phase inverter-fed balanced machine systems that is capable of calculating up to 24 harmonic components of the line currents every 130 mu s.
Abstract: A spectrum analyzer for three-phase inverter-fed balanced systems which is capable of calculating up to 24 harmonic components of the line currents every 130 mu s is presented. The method is based on a synchronized sampling technique and on a highly efficient fast Fourier transform (FFT) for three-phase systems. The latter consists of a two-dimensional six-point discrete Fourier transform (DFT) followed by a two-dimensional four-point DFT. The total FFT algorithm has been successfully implemented on a TMS32010 digital signal processor. >
01 Apr 1989
TL;DR: Algorithms for computing discrete Fourier transform (DFT) of 1-D real symmetric and antisymmetric sequences, using the prime-factor algorithm (PFA) and the Winograd Fourier Transform algorithm (WFTA), are presented.
Abstract: In the paper, algorithms for computing discrete Fourier transform (DFT) of 1-D real symmetric and antisymmetric sequences, using the prime-factor algorithm (PFA) and the Winograd Fourier transform algorithm (WFTA), are presented. These algorithms are obtained from different factorisations of the Fourier matrix, and it is shown that symmetry conditions exist at each stage which are used to construct efficient algorithms for computing DFTs.
03 Jan 1989
TL;DR: One of the Fast Fourier Transform algorithms, the Prime Factor algorithm (PFA), is implemented on the hypercube using a concurrent communication algorithm, called the Crystal_Router, to overcome the extra communication requirement up to a certain number of processors.
Abstract: We have implemented one of the Fast Fourier Transform algorithms, the Prime Factor algorithm (PFA), on the hypercube. On sequential computers, the PFA and other discrete Fourier transforms (DFT) such as the Winograd algorithm (WFA) are known to be very efficient. However, both algorithms require full data shuffling and are thus challenging to any distributed memory parallel computers. We use a concurrent communication algorithm, called the Crystal_Router for communicating shuffled data. We will show that the speed gained in reduced arithmetic compared to binary FFT is sufficient to overcome the extra communication requirement up to a certain number of processors. Beyond this point the standard Cooley-Tukey FFT algorithm has the best performance. We comment briefly on the application of the DFT to signal processing in synthetic aperture radar (SAR).
TL;DR: This paper presents a two dimensional FFT program (SW2DFFT), a Fortran program capable of handling large data matrices both square and rectangular, based on the decomposed Cooley-Tukey algorithm.
TL;DR: A fast algorithm for the arbitrary polynomial transformation is described, based on the fast Fourier transform algorithm, which reduces the computational complexity of a recently proposed recursive algorithm by an order of magnitude.
Abstract: A fast algorithm for the arbitrary polynomial transformation is described. This algorithm is based on the fast Fourier transform (FFT) algorithm and reduces the computational complexity of a recently proposed recursive algorithm by an order of magnitude. >
TL;DR: PFA algorithms are presented that take advantage of the symmetry in a real-even or real-odd sequence that require only one-fourth the real arithmetic and storage requirements of the complex PFA.
Abstract: The prime factor algorithm (PFA) is a fast algorithm for the evaluation of the discrete Fourier transform (DFT), applicable when the sequence length N is a product of relative primes. PFAs are presented that take advantage of the symmetry in a real-even or real-odd sequence. These algorithms require only one-fourth the real arithmetic and storage requirements of the complex PFA. As with existing state-of-the-art PFAs, these algorithms can be performed in-place and in-order.
26 Mar 1989
TL;DR: A multi-radix fast Fourier number theoretic transform is proposed for the calculation of the discrete Fourier transform of sequences with a prime length P=2/sup k1/*3/ Sup k2/*5/Sup k3/+1, where k1, k2, and k3 are integers.
Abstract: A multi-radix fast Fourier number theoretic transform is proposed for the calculation of the discrete Fourier transform of sequences with a prime length P=2/sup k1/*3/sup k2/*5/sup k3/+1, where k1, k2, and k3 are integers. Advantages include availability of fast algorithms for a set of prime lengths, residue arithmetic with benefit in speed and hardware cost, and parallel implementation. A discrete power spectrum example is included. >