Showing papers on "Prime-factor FFT algorithm published in 1986"

Phase-retrieval stagnation problems and solutions

Abstract: The iterative Fourier-transform algorithm has been demonstrated to be a practical method for reconstructing an object from the modulus of its Fourier transform (i.e., solving the problem of recovering phase from a single intensity measurement). In some circumstances the algorithm may stagnate. New methods are described that allow the algorithm to overcome three different modes of stagnation: those characterized by (1) twin images, (2) stripes, and (3) truncation of the image by the support constraint. Curious properties of Fourier transforms of images are also described: the zero reversal for the striped images and the relationship between the zero lines of the real and imaginary parts of the Fourier transform. A detailed description of the reconstruction method is given to aid those employing the iterative transform algorithm.

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.

On computing the split-radix FFT

Abstract: This paper presents an efficient Fortran program that computes the Duhamel-Hollmann split-radix FFT. An indexing scheme is used that gives a three-loop structure for the split-radix FFT that is very similar to the conventional Cooley-Tukey FFT. Both a decimation-in-frequency and a decimation-in-time program are presented. An arithmetic analysis is made to compare the operation count of the Cooley-Tukey FFT fo several different radixes to that of the split-radix FFT. The split-radix FFT seems to require the least total arithmetic of any power-of-two DFT algorithm.

Fast algorithms for the multidimensional discrete Fourier transform

Abstract: In this paper, the prime factor algorithm for the evaluation of a one-dimensional discrete Fourier transform is generalized to the evaluation of multidimensional discrete Fourier transforms defined on arbitrary periodic sampling lattices. It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform. As a sidelight to the derivation of the algorithm, a Chinese remainder theorem is derived for integer lattices.

Fast interpolation algorithm using FFT

Abstract: An interpolation algorithm for finite-duration real sequences using the discrete Fourier transform is presented. The proposed method is shown to result in a significant saving of computational labour over the discrete version of the time-domain classical interpolation formula. Estimation of inbetween samples for large sequences is possible within a mean square error of 0.00114 with this method. Some considerations with regard to the computation of FFTs are also discussed.

A new FFT algorithm of radix 3,6, and 12

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.

A VLSI chip for the winograd/Prime factor algorithm to compute the discrete Fourier transform

Abstract: There is an extensive literature about computing the discrete Fourier transform and the hardware implementations of the different algorithms. In this paper, we propose a radically different approach based on the so-called small n algorithms and several different iteration methods. Our approach will result in fully pipelined bit serial architectures which require no control units. The area is about the minimum possible, and the overall delay is within an optimal order magnitude. An essential ingredient of these implementations is the use of digit on-line adder and multiplier cells.

Pruning the decimation-in-time FFT algorithm with frequency shift

Abstract: Fourier transformed components within desired narrow-band can be efficiently calculated by the pruned version of the decimation-in-time FFT algorithm. A new pruning method is proposed here which invloves frequency shift. The shifting simplifies the pruning algorithm because its flowgraph has a repetitive pattern of butterflies between adjacent stages.

Vector-radix algorithm for a 2-D discrete Hartley transform

Abstract: A new multidimensional Hartley transform is defined and a vector-radix algorithm for fast computation of the transform is developed. The algorithm is shown to be faster (in terms of multiplication and addition count) compared to other related algorithms.

Conversion of FFT's to Fast Hartley Transforms

Abstract: The complex Fourier transform of a real function and its real Hartley transform are expressed in terms of each other, allowing translation of theorems and computer programs between the two versionsAny FFT can thus be converted, by a few indexing changes, into a Fast Hartley Transform which is equally efficient, in terms of floating point operations per real datum transformed The FHT can therefore transform one real array of length N in half the time that it takes the FFT to process a complex array of length N Several tricks to speed up both FHT and FFT are presented and a Fortran version of the FHT is supplied which delivers the result in $75\log _2 N$ multiplications and $175\log _2 N$ additions

Computing the complete fft of a step-like waveform

Abstract: Methods for duration limiting a step-like waveform for fast Fourier transform (FFT) computation are surveyed and discussed, and a complete FFT method is introduced. The complete FFT has an enhanced resolution, and is complete in the sense that it has uniformly spaced DC and harmonic components.

Performance of fixed-point FFT's: Rounding and scaling considerations

Abstract: The calculation of the discrete Fourier transform using a fast Fourier transform (FFT) algorithm with fixed-point arithmetic is considered. The input data is scaled to prevent overflow and to maintain accuracy. The implementation uses 16-bit fixed-point representation for the data and provides for double precision accumulation of sums and products. Algorithm variants as well as different rounding options are compared. Execution times for implementations based on a single chip signal processor are given. These show that a considerable increase in accuracy can be obtained with only a small penalty in execution time, by applying an alternating form of rounding rather than truncation.

Two and three dimensional FFTS on highly parallel computers

Abstract: An algorithm is described for computing two and three dimensional Fourier transforms on computers of SIMD architecture. The algorithm assumes the existence of a library routine for the calculation of a 2-d Fourier transform on a set of Np2 data points where Np2 is the number of processing elements. The paper discusses how to use this routine to calculate 2-d Fourier transforms on a set of N2 data points where NpN is a power of two, using an interleaving technique. The paper also discusses the use of the 2-d results in conjunction with base ‘r1 + r2’ FFT algorithms to calculate 3-d Fourier transforms on a set of N3 complex data points. In the final section a general program is described to calculate 3-d Fourier transforms for any values of N and Np such that NpN is a power of two. Timings are given for the algorithms run on an ICL Distributed Array Processor.

Easy generation of small-Ndiscrete Fourier transform algorithms

Abstract: In the paper, a set of rules is provided that allow construction of a wide range of efficient Rader's discrete Fourier transform (DFT) algorithms for the sizes of N being (a power of) an odd prime, having a limited set of polynomial reduction, multiplication and reconstruction algorithms. In theory, the algorithms obtained meet the lower bound on the number of multiplications. In practice, they have the same performance as the existing ones, and some new interesting algorithms are obtained, in particular for N = 25 and 27. For N being a power of 2, the use of the radix-4/2 fast Fourier transform is proposed, as this algorithm exhibits excellent properties when used as a small-N DFT algorithm. The paper contains two results of more general application: a rule determining when to use nesting or row-column algorithms, and propositions of new bounds on the number of multiplications for DFT algorithms for N = Pr, where p is a prime number.

Optimal algorithm for computing the two-dimensional discrete fourier transform.

Abstract: Since for each t ∈ [1, N/2], we have W t+N/2 = −W , one can also represent component (1) at the point (p, s) by the corresponding N/2-dimensional vector F̄ ′ p,s = (f ′ p,s,1, f ′ p,s,2, ..., f ′ p,s,N/2), whose components are calculated from the components of the corresponding initial vector F̄p,s by formula f ′ p,s,t = fp,s,t − fp,s,t+N/2, t = 1 ÷ N/2. (5) We call such representation of each component Fp,s of the spectrum in the form of the corresponding N/2-dimensional vector F̄ ′ p,s the paired vector representation, to distinct it from the original vector representation F̄p,s, and the constructed tensor of the 3rd order (f ′ p,s,t; p, s, = 1 ÷ N, t = 1 ÷ N/2 to be the paired tensor of the Fourier-spectrum. As for the original tensor representation of the spectrum of the signal, when for any p, s and k the following formula was valid [1]

Cyclic convolution of real sequences: Hartley versus Fourier and new schemes

Abstract: Recently, new fast transforms (such as the discrete Hartley transform in particular) have been proposed which are best suited for the computation of cyclic convolution of real sequences. Two approaches using Fourier or Hartley transforms are first compared, showing that the recently proposed FFT algorithms for real data present a lower arithmetic complexity than the corresponding DHT-based approach. Improvements are made to both types of algorithms, leading to different trade offs between arithmetic and structural complexity. We also present a new Hartley Transform algorithm with lower arithmetic complexity than any previously published one.

Fast Fourier transform data address pre-scrambler circuit

Abstract: A fast Fourier transform (FFT) data address pre-scrambler technique and cuit for selectively generating pre-scrambled bit reversed, data address sequences needed to perform radix-2, radix-4 and mixed radix-2/4 fast Fourier transforms.

Fast algorithms for the discrete Fourier transform and for other transforms

Abstract: A new matrix factorization is proposed for DCT-IV, which is the basis of fast algorithms for many sinusoidal transforms. A new fast algorithm for complex-data DFT based on the new factorization requires the same number of multiplications and far fewer additions than the Preuss algorithm. A new fast algorithm for real-data DFT based on a new algorithm for the discrete Hartley transform is also proposed.

An efficient vector implementation of the FFT algorithm on IBM 3090VF

Abstract: In this paper, an efficient vector implementation of the fast Fourier transform (FFT) algorithm on the IBM 3090 Vector Facility is presented. This is a part of the Engineering and and Scientific Subroutine Library (ESSL). The implementation works with the full vector length of the machine and the cache is also efficiently managed to achieve very good performance. For short length transforms, a multiple number of transforms could be computed to improve performance. The performance of the vector rountines is compared against state of the art scalar routines and the performance improvements of up to a factor of 8 are observed.

VLSI Implementation Of The Fast Fourier Transform

Abstract: A VLSI implementation of a Fast Fourier Transform (FFT) processor consisting of a mesh interconnection of complex floating-point butterfly units is presented. The Cooley-Tukey radix-2 Decimation-In-Frequency (DIF) formulation of the FFT was chosen since it offered the best overall compromise between the need for fast and efficient algorithmic computation and the need for a structure amenable to VLSI layout. Thus the VLSI implementation is modular, regular, expandable to various problem sizes and has a simple systolic flow of data and control. To evaluate the FFT architecture, VLSI area-time complexity concepts are used, but are now adapted to a complex floating-point number system rather than the usual integer ring representation. We show by our construction that the Thompson area-time optimum bound for the VLSI computation of an N-point FFT, area-time2oc = ORNlogN)1+a] can be attained by an alternative number representation, and hence the theoretical bound is a tight bound regardless of number system representation.

On a fast algorithm for the approximate solution of the discrete Wiener-Hopf equation and an estimation of the accuracy

Abstract: A fast algorithm for the approximate solution of the discrete Wiener-Hopf equation is described and convergence rate estimates are obtained. The numerical algorithm is constructed by using the discrete Fourier transformation.

A fast Fourier transform algorithm using Hadamard transform

Abstract: A fast Fourier transform (FT) algorithm using Hadamard transform (HT) is introduced, which is called HFT (Hadamard Fourier Transform). In the algorithm proposed here, a HT is used as mid-transform and the redundant calculation in the original fast FT algorithm is reduced by double transformation. The results of theoretical analysis show that the number of multiplications and additions of HFT are both decreased by 60% compared with that of traditional FFT and the executed result shows the computing speed of HFT is 1.6 to 1.7 times faster than FFT. Comparing with the similar algorithms such as WFT-II1, RFT2, it has a market improvement in computing speed and eliminates the limitatiom on the length of transform.

Correction of some formulas of Agarwal's fast Fourier transform least‐squares algorithm

Abstract: Correction of formulas (34), (37) and (38) of Agarwal's fast Fourier transform least-squares algorithm [Acta Cryst. (1978), A34, 791-809].

Efficient Implementation of Prime Factor Fourier Transforms

Abstract: The Winograd small factor Fourier transforms can be formed with a small number of common signal flow structures reminiscent of the butterfly in the Cooley Tuckey radix-2 fast Fourier transform (FFT) algorithm This paper identifies these structures and describes a versitile and compact prime factor algorithm (PFA) which incorporates them Running time and program code space requirements for this algorithm are presented and are compared to classic radix-2 and radix-4 FFT algorithms

EFFICIENTIMPLEMENTATION OF PRIMEFACTORFOURIERTRANSFORMS fredharris

Abstract: The Winogradsmallfactor Fourier transformscan be formedwith a smallnumberof commonsignalflowstructuresreminiscent of thebutterflyin theCooley Tuckey radix-2 fast Fouriertransform(FFT) algorithm.This paperidentifiesthesestructures and describesa versitileand compact prime factor algorithm(PFA) which incorporatesthem. Runningtime and program code space requirementsfor this algorithmare presentedand are compared to classicradix-2and radix-4FFT algorithms . BACKGROUND The discreteFouriertransform(DFT) of length N implementedin the directinner productor matrixformrequiresN2 arithmetic operations. When N is composite, thatis, the productof two factorsN1 and N2, the transformcan be implementedas one of the FastFourierTransform (FFT) algorithms.These algorithmsmap, by various reindexingschemes,the one dimensionalDFT of lengthN intoan N1by N2 two dimensionalDFT, The reindexingschemefor the CooleyTukey algorithm(CTA)is easyto implement(lexicographical) but results in a residual couplingbetweenthe transformsover the two dimensions.This coupling,a phase correctionterm called the twiddle factor,must be appliedto each datapointas a complexproductbetweentransformdirections. The most commonlyused formof thisalgorithm are variantsof the radix-2 and radix-4FFT.These variantsinclude rearrangement(ordecimation) in time and rearrangementin frequencyand in- place and non in- placeFFTsO. By iteratively mappingeachNm transformto an Nm/2 by 2 two dimensionaltransformthe computational workloadfor the radix-2(complexdata) transformis on the order of 2N log2(N) realmulttiplications

Recursive Transforms In Hybrid Processing

Abstract: This paper presents a unified analysis of a class of unitary transforms including the discrete Fourier, the Walsh Hadamard, the discrete Hartley, and the discrete cosine transforms. These transforms possess a common recursive property that allows us to obtain the next higher-order transform from two identical, preceding lower-order transforms. This recursive property eventually leads us to formulate a fast, efficient recursive algorithm for the discrete Hartley transform, from which the fast processing algorithm for the discrete cosine transform can also be obtained. Hybrid implementations using state-of-the-art integrated optics in digital format have also been proposed for such fast, efficient processing algorithms.

A VLSI Chip for the Winograd/Prime Factor Algorithm to Compute the Discrete Fourier Transform

Abstract: There is an extensive literature about computing the dis- crete Fourier transform and the hardware implementations of the dif- ferent algorithms In this paper, we propose a radically different ap- proach based on the so-called small n algorithms and several different iteration methods Our approach will result in fully pipelined bit serial architectures which require no control units The area is about the minimum possible, and the overall delay is within an optimal order magnitude An essential ingredient of these implementations is the use of digit on-line adder and multiplier cells