One-dimensional Fast Fourier Transform Research Articles

Spirality: A Novel Way to Measure Spiral Arm Pitch Angle

We present the MATLAB code Spirality, a novel method for measuring spiral arm pitch angles by fitting galaxy images to spiral templates of known pitch. Computation time is typically on the order of 2 min per galaxy, assuming 8 GB of working memory. We tested the code using 117 synthetic spiral images with known pitches, varying both the spiral properties and the input parameters. The code yielded correct results for all synthetic spirals with galaxy-like properties. We also compared the code’s results to two-dimensional Fast Fourier Transform (2DFFT) measurements for the sample of nearby galaxies defined by DMS PPak. Spirality’s error bars overlapped 2DFFT’s error bars for 26 of the 30 galaxies. The two methods’ agreement correlates strongly with galaxy radius in pixels and also with i-band magnitude, but not with redshift, a result that is consistent with at least some galaxies’ spiral structure being fully formed by z=1.2, beyond which there are few galaxies in our sample. The Spirality code package also includes GenSpiral, which produces FITS images of synthetic spirals, and SpiralArmCount, which uses a one-dimensional Fast Fourier Transform to count the spiral arms of a galaxy after its pitch is determined. All code is freely available.

Reducing the two-loop large-scale structure power spectrum to low-dimensional, radial integrals

Modeling the large-scale structure of the universe on nonlinear scales has the potential to substantially increase the science return of upcoming surveys by increasing the number of modes available for model comparisons. One way to achieve this is to model nonlinear scales perturbatively. Unfortunately, this involves high-dimensional loop integrals that are cumbersome to evaluate. Trying to simplify this, we show how two-loop (next-to-next-to-leading order) corrections to the density power spectrum can be reduced to low-dimensional, radial integrals. Many of those can be evaluated with a one-dimensional Fast Fourier Transform, which is significantly faster than the five-dimensional Monte-Carlo integrals that are needed otherwise. The general idea of this FFT-PT method is to switch between Fourier and position space to avoid convolutions and integrate over orientations, leaving only radial integrals. This reformulation is independent of the underlying shape of the initial linear density power spectrum and should easily accommodate features such as those from baryonic acoustic oscillations. We also discuss how to account for halo bias and redshift space distortions.

Modification of a two-dimensional fast Fourier transform algorithm by the analog of the Cooley-Tukey algorithm for a rectangular signal

One-dimensional fast Fourier transform (FFT) is the most popular tool for computing the two-dimensional Fourier transform. As a rule, a standard method of combination of one-dimensional FFTs--the so-called algorithm "by rows and columns" [1]--is used in the literature. In [2, 3], the authors showed how to compute the FFT for a signal with the number of samples 2 s × 2 s with the use of an analog of the Cooley-Tukey algorithm. In the present paper, a two-dimensional analog of the Cooley-Tukey algorithm is constructed for a rectangular signal with the number of samples 2 s × 2 s + l. The number of operations in this algorithm is much less than that in the successive application of a one dimensional FFT by rows and columns. The testing of the algorithm on image-type signals shows that the speed of computation of the FFT by the algorithm proposed is about 1.7 times higher than that of the algorithm by rows and columns.

Calculating the n-dimensional fast Fourier transform

The one-dimensional fast Fourier transform (FFT) is the most popular tool for calculating the multidimensional Fourier transform. As a rule, to estimate the n-dimensional FFT, a standard method of combining one-dimensional FFTs, the so-called "by rows and columns" algorithm, is used in the literature. For fast calculations, different researchers try to use parallel calculation tools, the most successful of which are searches for the algorithms related to the computing device architecture: cluster, video card, GPU, etc. [1, 2]. The possibility of paralleling another algorithm for FFT calculation, which is an n-dimensional analog of the Cooley-Tukey algorithm [3, 4], is studied in this paper. The focus is on studying the analog of the Cooley-Tukey algorithm because the number of operations applied to calculate the n-dimensional FFT is considerably less than in the conventional algorithm nN n log2 N of addition operations and 1/2N n + 1log2 N of multiplication operations of addition operations and $$\frac{{2^n - 1}} {{2^n }}N^n \log _2 N$$ of multiplication operations against: N n + 1log2 N of addition operations and 1/2N n + 1log2 N of in combining one-dimensional FFTs.

Large-scale fast Fourier transform on a heterogeneous multi-core system

As interest in hybrid computing systems increases, people are eager to find new ways to exploit the unique and efficient computational power of the heterogeneous multi-core systems. Although there has been much interest in implementing high-performance fast Fourier transform (FFT) libraries on this kind of system, most existing libraries focus on small-scale FFTs whose data can fit in the local storage of a single accelerator. Real-world FFT applications often require much larger scale FFTs, but it is extremely challenging for heterogeneous multi-core system with distributed architectures to make efficient large FFT implementations. In this paper, we introduce the first known FFT library for the heterogeneous multi-core system with distributed architecture that can solve one-dimensional FFTs larger than what fits in a single accelerator. Our implementation achieves 67% performance improvement of FFTW 3.2.2 (Fastest Fourier Transform in the West) and sustains over 36 single precision FFT GFLOPs ‘end-to-end’ Achieving such high performance requires novel schemes for large-scale FFT factorization, data permutation and all-to-all exchanges, and buffer designs to maximize use of the local storage while minimizing communication overhead. One important finding in this paper is that large-scale FFT on this kind of architecture behaves as data transfer bound which is quite different from other architectures. A significant contribution of this paper is that for each major component of our algorithm, we explore many possible design options and present quantitative performance comparisons. This provides value beyond specific architecture, as it illustrates the fundamental features associated with different communication paradigms and mechanisms for the heterogeneous multi-core system. Today’s computer systems are increasingly being designed to include general purpose accelerators. The techniques in this paper can also be applied to these architectures especially when they have limited local storage or steep cache hierarchies. We also provide insights on applying techniques in this paper to similar architectures.

A parallel 1-D FFT algorithm for the Hitachi SR8000

In this paper, we propose a high-performance parallel one-dimensional fast Fourier transform (FFT) algorithm on clusters of vector symmetric multiprocessor (SMP) nodes. The four-step FFT algorithm can be altered into a five-step FFT algorithm to expand the innermost loop length. We use the five-step algorithm to implement the parallel one-dimensional FFT algorithm. In our proposed parallel FFT algorithm, since we use cyclic distribution, all-to-all communication takes place only once. Moreover, the input data and output data are both in natural order. Performance results of one-dimensional power-of-two FFTs on clusters of pseudo-vector SMP nodes, Hitachi SR8000, are reported. We succeeded in obtaining performance of over 61 GFLOPS on a 16-node Hitachi SR8000/MPP.

FAR-ZONE CONTRIBUTIONS TO TOPOGRAPHICAL EFFECTS IN THE STOKES-HELMERT METHOD OF THE GEOID DETERMINATION

In the evaluation of the geoid done according to the Stokes-Helmert method, the following topographical effects have to be computed: the direct topographical effect, the primary indirect topographical effect and the secondary indirect topographical effect. These effects have to be computed through integration over the surface of the earth. The integration is usually split into integration over an area immediately adjacent to the point of interest, called the near zone, and the integration over the rest of the world, called the far zone. It has been shown in the papers by Martinec and Vanicek (1994), and by Novak et al. (1999) that the far-zone contributions to the topographical effects are, even for quite extensive near zones, not negligible. Various numerical approaches can be applied to compute the far-zone contributions to topographical effects. A spectral form of solution was employed in the paper by Novak et al. (2001). In the paper by Smith (2002), the one-dimensional Fast Fourier Transform was introduced to solve the problem in the spatial domain. In this paper we use two-dimensional numerical integration. The expressions for the far-zone contributions to topographical effects on potential and on gravitational attraction are described, and numerical values encountered over the territory of Canada are shown in this paper.

An improved gravimetric geoid for Japan, JGEOID98, and relationships to marine gravity data

A geoid model for Japan, JGEOID98, is determined on a 3 × 3 arc-minute grid in the non-tidal system by a one-dimensional Fast Fourier Transform method with EGM96 as a foundation. The model is referred to the GRS80 ellipsoid/ITRF89 frame. About 570 000 ship gravity data are adjusted for about 50 000 cross-over errors (COEs) and yield significant improvement: root mean square of internal and external COEs after adjustment are 6.8 and 10.4 mGal (0 and 24% of reduction), respectively. Compared with GPS at benchmarks, JGEOID98 shows improvement over JGEOID93 by 29% in short wavelength, and tilt reduction from 2.1 to 1.4 ppm. The remaining trend can be partly explained by gravity errors: overestimation of gravity induced from EGM96 east off Hokkaido, and underestimation south off Taiwan.

Plane-wave scattering-matrix formulation for two interfaces plus a scatterer (uniplanar case)

Kern's plane-wave scattering-matrix formulation is extended to treat the case of two interfaces plus a scatterer imbedded in the second region. This formulation can treat different shapes for the scatterer and only requires two different evaluations depending upon the location of the observer: 1) in the near field, a two-dimensional (2-D) fast Fourier transform (FFT) (one-dimensional (1-D) FFT is treated by an analytical integration), and 2) in the far-field an asymptotic evaluation of the integral. These two regimes are in contrast to the Sommerfeld integral approximations where different approximations are required for various parameter ranges (quasistatic, saddle-point evaluation for the radiation field, saddle point evaluation for the surface field when the saddle point is near a pole, and the lateral wave field evaluated using a uniform asymptotic evaluation for the integrals).

Performing out-of-core FFTs on parallel disk systems

The Fast Fourier Transform (FFT) plays a key role in many areas of computational science and engineering. Although most one-dimensional FFT problems can be solved entirely in main memory, some important classes of applications require out-of-core techniques. For these, use of parallel I/O systems can improve performance considerably. This paper shows how to perform one-dimensional FFTs using a parallel disk system with independent disk accesses. We present both analytical and experimental results for performing out-of-core FFTs in two ways: using traditional virtual memory with demand paging, and using a provably asymptotically optimal algorithm for the Parallel Disk Model (PDM) of Vitter and Shriver. When run on a DEC 2100 server with a large memory and eight parallel disks, the optimal algorithm for the PDM runs up to 144.7 times faster than in-core methods under demand paging. Moreover, even including I/O costs, the normalized times for the optimal PDM algorithm are competitive, or better than, those for in-core methods even when they run entirely in memory.

Recursive pruning of the 2D DFT with 3D signal processing applications

A recursively pruned radix-(2*2) two-dimensional (2D) fast Fourier transform (FFT) algorithm is proposed to reduce the number of operations involved in the calculation of the 2D discrete Fourier transform (DFT). It is able to compute input and output data points having multiple and possibly irregularly shaped (nonsquare) regions of support. The computational performance of the recursively pruned radix-(2*2) 2D FFT algorithm is compared with that of pruning algorithms based on the one-dimensional (1D) FFT. The former is shown to yield significant computational savings when employed in the combined 2D DFT/1D linear difference equation filter method to enhance three-dimensional spatially planar image sequences, and when employed in the MixeD moving object detection and trajectory estimation algorithm. >

Computing discrete Fourier transform on a rectangular data array

The authors generalize the weighted redundancy transform (WRT) algorithm for computing the multidimensional discrete Fourier transform (DFT) in the case in which the sample size (blocklength) is not the same on every axis. The proposed algorithm, like the WRT algorithm, is based on the one-dimensional fast Fourier transform (FFT) and, compared to the traditional ways of computing the multidimensional DFT, offers substantial savings in the number of one-dimensional FFT procedure calls. While the algorithm is applicable to transforms of any dimensions, only the two-dimensional case is explored in detail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multidimensional fast Fourier transform into SIMD hypercubes

The paper presents the in-place implementation of the multidimensional radix 2 fast Fourier transform (FFT), along with the corresponding algorithm for data shuffling (bit-reversal) on SIMD hypercube computers. Each processor possesses its own non-shared memory, the number of processors being less than or equal to the number of data. The flexibility of the proposed algorithm is based on the scheme of information storage that has been chosen and in the decomposition/configuration of the hypercube in subhypercubes that allow the parallel processing of multiple one-dimensional FFTs. This parallel FFT algorithm has an optimum performance, since the data redundancy is null and the algorithmic complexity is optimum. >

A combined finite element-boundary integral formulation for solution of two-dimensional scattering problems via CGFFT

A novel technique is presented for computing the scattering by two-dimensional structures of arbitrary inhomogeneity. The proposed approach combines the usual finite-element (FE) method with the boundary-integral equation to formulate a discrete system. This is subsequently solved via the conjugate gradient (CG) algorithm. A particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer. Several of the resulting boundary integrals are then convolutions and can be evaluated via the fast Fourier transform (FFT) in the implementation of the CG algorithm. The solution approach presented here offers the principal advantage of having O(N) memory demand and employs a one-dimensional FFT, as against the two-dimensional FFT required in a traditional implementation of the proposed CG-FFT algorithm. The speed of the proposed solution method is compared with that of the traditional CG-FFT algorithm. Results are presented for several rectangular composite cylinders and one perfectly conducting cylinder. These are shown to be in excellent agreement with the moment method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Gravity effect of axially symmetric bodies using the one-dimensional FFT

Summary. A technique for rapid calculation of the vertical gravitational attraction of three-dimensional axially symmetric bodies using the onedimensional Fast Fourier Transform (FFT) is presented and discussed. The calculation is rapid, efficient and accurate, and requires only a simple linear array to describe the body shape. This method is most useful for calculation of the gravity effect of long-wavelength bodies such as oceanic seamounts or sedimentary basins and for the gravity effect of annular bodies such as upwarped sedimentary strata around a salt diapir.

Roundoff error in multidimensional generalized discrete transforms

The analysis of rounding error in the one-dimensional fast Fourier transform (FFT) is extended to a class of generalized orthogonal transforms [1] with a common fast algorithm similar to the FFT algorithm. This class includes the BInary FOurier REpresentation (BIFORE) transform (BT) [2], the complex BT (CBT) [3], and the discrete Fourier transform (DFT). Expressions for the mean square error (MSE) in the two-dimensional BT, CBT, and FFT are derived. In the case of white input data, the mean square error-to-signal ratio is derived for the multidimensional generalized transforms. The error-to-signal ratio for the one-dimensional FFT derived by Kaneko and Liu is modified with improvement. Some comparisons among BIFORE, DFT, and Haar transforms are also included. The theoretical results for the two-dimensional FFT and BIFORE have been verified experimentally. The experimental results are in good agreement with the theoretical results for lower order sequences, but deviate as the order increases due to the actual manner of rounding in the digital computer.