New radix-3 and −6 decimation-in-frequency fast Hartley transform algorithms
01 Apr 1993-Canadian Journal of Electrical and Computer Engineering-revue Canadienne De Genie Electrique Et Informatique (IEEE)-Vol. 18, Iss: 2, pp 65-69
TL;DR: From the results, it is seen that the radix-3 and -6 FHT algorithms presented are comparable to the split-radix FHT algorithm in terms of their operation count and will be more efficient when the sequence length is closer to an integer power of the corresponding radix.
Abstract: Fast algorithms of a transform, like fast Fourier transform (FFT) algorithms, are based on different decomposition techniques. It is shown that these decomposition techniques can also be applied to the computation of the discrete Hartley transform (DHT) for a real-valued sequence. Recently, an efficient decomposition technique for radix-3 decimation-in-time (DIT) FFT and fast Hartley transform (FHT) algorithms has been demonstrated. Such a decomposition technique is implemented for radix-3 and -6 decimation-in-frequency (DIF) FHT algorithms and found to improve the operation count. Efficiency in these algorithms is derived by pairing the rotating factors with an appropriate reordering of the input sequence. From the results, it is seen that the radix-3 and -6 FHT algorithms presented are comparable to the split-radix FHT algorithm in terms of their operation count and will be more efficient when the sequence length is closer to an integer power of the corresponding radix.
Citations
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23 Jul 2009
TL;DR: A new fast radix-2 decimation-in-frequency algorithm for computing the DHT that requires less number of multiplications than those presented by Bracewell, Meckelburg and Lipka, Prado and Sorenson et al is proposed.
Abstract: The radix-2 decimation-in-time fast Hartley transform algorithm for computing the Discrete Hartley Transform (DHT) was introduced by Bracewell. A radix-2 decimation-in-frequency algorithm by Meckelburg and Lipka followed. Prado came up with an in-place version of Bracewells decimation-in-time fast Hartley transform algorithm. A set of fast algorithms for both decimation-in-time and decimation-in-frequency was further developed by Sorenson et al. A new fast radix-2 decimation-in-frequency algorithm for computing the DHT that requires less number of multiplications than those presented by Bracewell, Meckelburg and Lipka, Prado and Sorenson et al is proposed. It exploits the characteristics of the DHT matrix, exhibits stage structures with butterflies similar for each stage and introduces multiplying structures in the signal flow diagram. The operation count for the proposed algorithm is determined. It is verified by implementing the program in C.
3 citations
Cites methods from "New radix-3 and −6 decimation-in-fr..."
...Several algorithms have been reported in the literature for its fast computation [5]-[12]....
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01 Nov 2008
TL;DR: A position-based method is proposed to simplify the computation of the elements of the discrete Hartley transform matrix and is faster in computing the elements than the direct method based on definition.
Abstract: The elements of the discrete Hartley transform matrix can be computed using the direct method in which each element of the matrix is calculated based on its definition. A position-based method is proposed to simplify the computation. In this method, the characteristics of the discrete Hartley transform matrix are identified and made use of to directly assign values to some elements and compute only a few typical distinct magnitude elements using its definition. These elements are then utilized to obtain the remaining elements based on their positions. An algorithm utilizing this position-based method is developed which is faster in computing the elements than the direct method based on definition.
2 citations
01 Jan 2014
TL;DR: The new Proposed design has been designed and developed by novel parallel-pipelined Fast Fourier Transform (FFT) architecture and the radix-6 Z has been developed to reduce the complexity of hardware and the computational intension.
Abstract: The new Proposed design has been designed and developed by novel parallel-pipelined Fast Fourier Transform (FFT) architecture. The most important and fastest efficient algorithm is a FFT. FFT is used to computes the Discrete Fourier Transform (DFT). FFT is mainly applied in autocorrelation, spectrum analysis, linear filtering and pattern recognition system. The proposed architectures were designed by using register minimization and folding transformation technique. The critical path is reduced by pipelining and the multiple inputs and multiple outputs are computed by parallel processing. Furthermore, radix-2, radix- 3, radix-6 Z decimation in time and decimation in frequency algorithm can be used. As a result, the radix-6 Z has been developed to reduce the complexity of hardware and the computational intension. The power dissipation can be decreased and unwanted operations are stopped by using the clock gates.
Cites background from "New radix-3 and −6 decimation-in-fr..."
...Later, the designers have found the new design algorithms for length N=3 m , that can be expanded by radix-3, radix-6, radix-9 [11]-[13]....
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01 Jan 2015
TL;DR: The new proposed algorithm for computing a length L=2 B x3 C FFT is used to minimize the twist of hardware depict and arithmetic operations and to belittle the area of system on chips and mathematical calculation.
Abstract: Fast Fourier Transform (FFT) is a most proficient algorithm for computing the Discrete Fourier Transform (DFT). FFT is specially used in analysis of autocorrelation, disambiguation, recognition of patterns, statistics and data analysis. The new proposed algorithm for computing a length L=2 B x3 C FFT. It is used to minimize the twist of hardware depict and arithmetic operations. Additionally, the propound fragment design can accomplish by the mixer of radix-3 and radix-2 B x3 C FFT algorithm. It is entirely changeable of Split Radix Fast Fourier Transform (SRFFT) algorithm. As a consequence, the new proffer design of length L is used to belittle the area of system on chips and mathematical calculation. More compromise can achieve by combining the two architectures implemented in the same FFT. It naturally furnishes a broad selection of accessible length of FFT’s.
References
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TL;DR: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable and promises to speed up Fourier-transform calculations.
Abstract: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ HartleyR. V. L., Proc. IRE30, 144 ( 1942)] as the DFT bears to the Fourier transform.
465 citations
"New radix-3 and −6 decimation-in-fr..." refers methods in this paper
...Since Bracewell's discovery of the Hartley transform [1] and the fast Hartley transform [2], considerable effort has gone into the devel opment of FHT algorithms....
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01 Aug 1984
TL;DR: The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied.
Abstract: A fast algorithm has been worked out for performing the Discrete Hartley Transform (DHT) of a data sequence of N elements in a time proportional to Nlog 2 N. The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied. A new timing diagram (stripe diagram) is presented to illustrate the overall dependence of running time on the subroutines composing one implementation; this mode of presentation supplements the simple counting of multiplies and adds. One may view the Fast Hartley procedure as a sequence of matrix operations on the data and thus as constituting a new factorization of the DFT matrix operator; this factorization is presented. The FHT computes convolutions and power spectra distinctly faster than the FFT.
455 citations
"New radix-3 and −6 decimation-in-fr..." refers methods in this paper
...Since Bracewell's discovery of the Hartley transform [1] and the fast Hartley transform [2], considerable effort has gone into the devel opment of FHT algorithms....
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TL;DR: A complete set of fast algorithms for computing the discrete Hartley transform is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms.
Abstract: The discrete Hartley transform (DHT) is a real-valued transform closely related to the DFT of a real-valued sequence. Bracewell has recently demonstrated a radix-2 decimation-in-time fast Hartley transform (FHT) algorithm. In this paper a complete set of fast algorithms for computing the DHT is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms. The philosophies of all common FFT algorithms are shown to be equally applicable to the computation of the DHT, and the FHT algorithms closely resemble their FFT counterparts. The operation counts for the FHT algorithms are determined and compared to the counts for corresponding real-valued FFT algorithms. The FHT algorithms are shown to always require the same number of multiplications, the same storage, and a few more additions than the real-valued FFT algorithms. Even though computation of the FHT takes more operations, in some situations the inherently real-valued nature of the discrete Hartley transform may justify this extra cost.
275 citations
"New radix-3 and −6 decimation-in-fr..." refers background in this paper
...[3] derived radix-2, radix4, split-radix, prime factor and Winograd FHT algorithms which have the optimal number of multiplications and additions....
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TL;DR: Through use of the fast Hartley transform, discrete cosine transforms (DCT) and discrete Fourier transforms (DFT) can be obtained and the recursive nature of the FHT algorithm derived in this paper enables us to generate the next higher order FHT from two identical lower order F HT's.
Abstract: The fast Hartley transform (FHT) is similar to the Cooley-Tukey fast Fourier transform (FFT) but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the FFT. Through use of the FHT, discrete cosine transforms (DCT) and discrete Fourier transforms (DFT) can be obtained. The recursive nature of the FHT algorithm derived in this paper enables us to generate the next higher order FHT from two identical lower order FHT's. In practice, this recursive relationship offers flexibility in programming different sizes of transforms, while the orderly structure of its signal flow-graphs indicates an ease of implementation in VLSI.
175 citations
"New radix-3 and −6 decimation-in-fr..." refers methods in this paper
...Now if, instead of H[k]9 0 < k < 5, we evaluate H[k], -2<k<3, where H[-\] = H[5] and H[-2] = H[4], we can pair the rotating factors....
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...The recursive nature of the FHT algorithms has been introduced by Hou [5]....
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TL;DR: A new algorithm for implementation of radix 3, 6, and 12 FFT is introduced, derived from the fact that, if an input sequence is favorably reordered, rotating factors can be treated in pairs so that the rotating factors are conjugate to each other.
Abstract: A new algorithm for implementation of radix 3, 6, and 12 FFT is introduced. An FFT using this algorithm is computed in an ordinary (1,j) complex plane and the number of additions can be significantly reduced; the number of multiplication is also reduced. High efficiency of the algorithm is derived from the fact that, if an input sequence is favorably reordered, rotating factors can be treated in pairs so that the rotating factors are conjugate to each other.
52 citations
"New radix-3 and −6 decimation-in-fr..." refers background in this paper
...Because the efficiency of these algorithms is based on pairing the rotating factors [7], indexing is not straightforward....
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