Standard Fast Fourier Transform Research Articles

Derivation and implementation of an efficient fast fourier transform algorithm (EFFT)

Abstract An analysis of the sequence of calculations during a standard radix-2 Fast Fourier Transform (FFT) computer algorithm reveals that additional multiplications beyond those normally eliminated by the conventional algorithms are also found to be unnecessary. In particular, the arguments for the trigonometric functions are always used in a well-defined “modified” bit-inverted order. The first four arguments within any pass yield reductions of multiplications via trigonometric identities. Further, by placing the trigonometric parameters in a look-up table in the same modified order, the minimum number of multiplications for a radix-2 FFT is performed by a simplified algorithm termed the EFFT (Efficient Fast Fourier Transform). This algorithm requires no reference to a bit-inversion procedure or look-up table until the output points are “unscrambled” at the conclusion. The feasibility of performing reasonably-large sized EFFT's on personal computers is examined. It is shown that using standard programming practices and currently available hardware, the execution time for the EFFT may he reduced to few seconds for complex transforms up to 2048 double-precision points. On an IBM Personal Computer, this transform may be performed in as little as 4.4 s.

Pi 2 pulsations and the eastward electrojet: A case study

We report the results of a case study of two Pi 2 pulsations observed near the eastward electrojet by the Scandinavian Magnetometer Array. The power of the two Pi 2 pulsations, calculated using a standard Fast Fourier Transform method, peaks near the centre of the eastward electrojet. For both events there is a strong latitudinal gradient in the power poleward of the equatorward border of the electrojet. The sense of polarisation is predominantly clockwise at the northern stations and anticlockwise at the southern stations although the reversal from clockwise to anticlockwise does not occur at a constant latitude. For the first event the polarisation reversal occurs at higher latitudes in the western half of the array; for the second the polarisation reversal occurs at higher latitudes at the edges of the array. The polarisation reversal does not appear to be related to the location of the eastward electrojet. Equivalent current vectors of the Pi 2 pulsations, obtained by rotating the band pass filtered data through 90°, exhibit clear vortex structures in both events. The vortices change sense of direction at half the period of the Pi 2 pulsation. A simple model for the ionospheric electric field in accord with the field line resonance theory reconstructs the basic features of the observed Pi 2 equivalent current system. We thus conclude that Pi 2 signatures in the region of the eastward electrojet and far away from the auroral break-up region are governed by the field line resonance mechanism.

Molecular wave functions in momentum space

Momentum-space calculations exhibit two kinds of advantages over position space: First, the numerical solution of Hartree-Fock equation is feasible without expansion of the wave functions in a particular basis. Equations only exhibit one avoidable singularity even for the multicenter case. Several mathematical techniques are presented, including standard fast Fourier-transform (FFT) techniques and numerical calculation of the involved convolutions. Second, momentum representation contributes in an original way to a better understanding of several physical problems arising in quantum chemistry. The two-body density matrix involving the electronic correlation are examined in both position and momentum space. If an expansion in Gaussian functions is used, momentum space renders feasible the obtainment of a multidimensional fully correlated wave function, starting from the Hartree-Fock solution.

Image Processing by Transforms Over a Finite Field

A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.

Fourier transform algorithms for interferometry

Several algorithms for performing Fourier transforms are discussed. After pointing out the advantages of the fast Fourier transform (FFT) algorithm it is shown that most of this algorithm is redundant for interferometric applications where the input is real and symmetric. For this case a new algorithm is described in detail (including a flowdiagram) which provides savings of core storage and execution time of a factor of four as compared to the standard FFT algorithm.