Showing papers on "Split-radix FFT algorithm published in 1985"

On computing the discrete Hartley transform

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.

Implementation of "Split-radix" FFT algorithms for complex, real, and real symmetric data

Abstract: A new algorithm is presented for the fast computation of the Discrete Fourier Transform. This algorithm belongs to that class of recently proposed 2n-FFT's which present the same arithmetic complexity (the lowest among any previously published one). Moreover, this algorithm has the advantage of being performed "in-place", by repetitive use of a "butterfly"- type structure, without any data reordering inside the algorithm. Furthermore, it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm.

A comparison of the time involved in computing fast Hartley and fast Fourier transforms

Abstract: It is shown that the DFT of a real sequence, formed via the Fast Hartley Transform, can be computed at most only 2 times faster than by using a complex Fast Fourier Transform. However, more sophisticated FFT algorithms exist which give the same speedup factor. A simple FHT subroutine is presented to illustrate the similarity of the FHT and FFT butterflies in their simplest forms.

The Fast Fourier Transform (FFT)

Abstract: The Fast Fourier Transform (FFT) is one of the most frequently used mathematical tools for digital signal processing. Techniques that use a combination of digital and analogue approaches have been increasing in numbers. This chapter is for establishing the basis of this combined approach in dealing with computer tomography, computer holography and hologram matrix radar.

Systolic implementations for deconvolution, DFT and FFT

Abstract: The paper presents a number of systolic configurations for computing deconvolutions and discrete Fourier transformations. Two approaches to deconvolution are considered: a time-domain approach, which is based on a systolic inversion of an associated Toeplitz matrix, generated by a wavefront propagation of the known system response, while the other approach, which is in the frequency domain, utilises systolic discrete Fourier transform (DFT) and fast Fourier transform (FFT) processors. The latter employs a systolic elevator concept, which circumvents the traditional need for global communications in the FFT. Aspects of hardware implementation and speed trade-offs are also discussed.

A new discrete Fourier transform algorithm using butterfly structure fast convolution

Abstract: This paper proposes a new approach to computing the discrete Fourier transform (DFT) with the power of 2 length using the butterfly structure number theoretic transform (NTT). An algorithm breaking down the DFT matrix into circular matrices with the power of 2 size is newly introduced. The fast circular convolution, which is implemented by the NTT based on the butterfly structure, can provide significant reductions in the number of computations, as well as a simple and regular structure, The proposed algorithm can be successively implemented following a simple flowchart using the reduced size submatrices. Multiplicative complexity is reduced to about 21 percent of that by the classical FFT algorithm, preserving almost the same number of additions.

Bit-level systolic array implementation of the winograd fourier transform algorithm

Abstract: A bit level systolic array system is proposed for the Winograd Fourier transform algorithm. The design uses bit-serial arithmetic and, in common with other systolic arrays, features nearest-neighbour interconnections, regularity and high throughput. The short interconnections in this method contrast favourably with the long interconnections between butterflies required in the FFT. The structure is well suited to VLSI implementations. It is demonstrated how long transforms can be implemented with components designed to perform a short length transform. These components build into longer transforms preserving the regularity and structure of the short length transform design.

A prime factor FFT algorithm with real valued arithmetic

Abstract: A prime factor FFT algorithm involving only real valued arithmetic is devised to compute the discrete Fourier transform of a real sequence. This letter extends an approach proposed by Bracewell.

Multiply/Add tradeoffs in length-2 n FFT algorithms

Abstract: The multiplicative complexity of the length-2ndiscrete Fourier Transform is derived. A constructive approach is followed which describes the set of algorithms that realize the minimum number of multiplications. The operation counts of minimum multiply algorithms are compared to other FFT algorithms. The basic method is extended to derive the multiplicative complexity of length-pnDFFs where p is an odd prime number.

The Fast Hartley Transform

Abstract: The Fast Hartley TransformH. S. HouElectronics and Optics Division, The Aerospace Corporation2350 E. El Segundo Blvd., El Segundo, California 90245AbstractThe Fast Hartley Transform (FHT) is similar to the Cooley -Tukey Fast Fourier Transform(FFT) but performs much faster because it requires only real arithmetic computationscompared to the complex arithmetic computations required by the FFT. Through use of theFHT, 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 thenext higher -order FHT from two identical lower -order FHTs. In practice, this recursiverelationship offers flexibility in programming different sizes of transforms, while theorderly structure of its signal flow graphs indicates an ease of implementation in VSLI.IntroductionRecently, Bracewe111,2 introduced the Discrete Hartley Transform (DHT) as a new member ofthe transform family. The DHT uses the real variable cos(2,rkn /N) + sin(2nkn /N) as thetransform kernel, while the Discrete Fourier Transform (DFT) uses the complex exponential,Exp(i2n kn /N), as the transform kernel. Thus, the DHT is intuitively simpler and hence,faster than the Fast Fourier Transform (FFT) since the multiplication of two complex varia-bles is equivalent to four real multiplications, and a complex addition is two ie.§.l addi-

Comments on and extensions to "Evaluation of Fourier integrals using a FFT with improved accuracy and its applications"

Abstract: The evaluation of Fourier integrals using fast Fourier transform (FFT) employing the Narasimhan scheme is examined. It is observed to be a special case of spline interpolation. His method is effective when the sampling rate is sufficiently high and piecewise linear approximation between sampling points is adequate. If knowledge about the signal smoothness is available, further improvement in accuracy can be achieved by judicious choice of high degree spine interpolant.

A fast pipelined DFT processor and its programming consideration

Abstract: A new discrete Fourier transform (DFT) processor with a pipelined structure has been developed. This processor is designed to optimise computation of the pair of operationsAx0 ±Bx1, which is mostly encountered in various fast DFT algorithms. For real-valued data and coefficients, the processor needs only two machine cycles to calculate the pair of operations. A straightforward multiple-stage transform algorithm has been proposed to implement real-valued prime-factor or radix-type transforms. About half of the computation can be saved by taking into account the fact that transform outputs are conjugate pairs for real inputs. The short Winograd Fourier transform algorithm is suggested as a basic building block for large transforms because it is more efficient than the fast Fourier transform.

Use of the FFT to Speed Analysis of Planar Symmetrical 3- and 5-Ports by the Integral Equation Method

Abstract: In a recent paper, it was shown that, for planar two-dimensional problems with symmetry, linear eigenvalue-impedence matrix entry relations may be used to simplify the integral equation method of analysis. In this paper, it is pointed out that, in the case of planar circuits with N-fold rotational symmetry, these linear relations take the form of the discrete Fourier transform (DFT). Consequently, the fast Fourier transform (FFT) may be used in its place to give a further substantial improvement in computational speed.

A microprogrammable peripheral unit and the FFT

Abstract: A circuit based on the AM2901 bit slice processor and a serial/parallel multiplier is described along with the algorithm to compute the base-2 DIF fast Fourier transform.

FFT Drithmetic element built on VLSI high performance gate array

Abstract: The Fast Fourier Transform (FFT) radix-2 butterfly is implemented on a single high speed bipolar gate array. Consideration has been given to questions of algorithm performance and architectural flexibility to achieve both high speed and low noise operation. Multiple FFT AE devices may be connected to form parallel array systems with high processing rates. Four system architectures utilizing the FFT chip employ one, \log_{2}N, N/2 , and (N/2)\log_{2} N arithmetic elements, respectively, for a transform of N points.