Prime-factor FFT algorithm

About: Prime-factor FFT algorithm is a research topic. Over the lifetime, 2346 publications have been published within this topic receiving 65147 citations. The topic is also known as: Prime Factor Algorithm.

Comparison of Four Fast Fourier Transform Algorithms

Abstract: : Comparisons of four FFT (Fast Fourier Transform) algorithms (Brenner's, Cooley's, Fisher's, and Singleton's) have been made on the basis of program execution time, storage, and accuracy. Major modifications have been made in the generation of the trigonometric values in the Cooley and Fisher algorithms, with significant improvements in accuracy. Entry of constants in all algorithms has been changed: the constants are approximated by the best binary representation for the UNIVAC 1108 computer. Three waveform examples are used in the comparisons, namely, linear FM, random numbers, and a unit ramp. Also, the sizes of the FFT's considered are limited to powers of 2, from 16 through 8192. The results indicate that Singleton's and Brenner's algorithms have the shortest execution times and occupy the least amount of computer storage, whereas Cooley's and Fisher's algorithms are the most accurate. For example, for an FFT of size 1024 on the linear FM waveform, the maximum relative errors for the four algorithms are 0.17 x 10 to the -6th power, 0.63 x 10 to the -7th power, 0.64 x 10 to the -7th power, 0.41 x 10 to the -5th power, respectively. Thus, there is no single best algorithm for all three criteria considered; rather, each algorithm has its own area of most effective applicability.

System Identification Using Fast Fourier Transform

Abstract: Algorithms for system identification applying throughout Fast Fourier Transform (FFT) to the major calculating operations are introduced. It is shown that by using data of about as twice length as system settling time and by truncating the incorrect correlation functions resulting from them, errors owing to finiteness of data can be avoided. It is shown that so as to suppress the effects owing to statistical fluctuation of input data or output noise, superposition of data in frequency domain is effective, and also the damping terms of poles or zeros can be efficiently evaluated by utilizing the phase change of the spectra of the impulse response sequence. The proposed method can be efficiently applied to relatively higher order systems or relatively rapidly time-variant systems because of high accuracy and high speed processing of FFT. Moreover, it needs not to assume the order of the system a priori, and yields a reasonable lower order approximating system in itself.

A Transforming Method of Hilbert Based on the Windowed Interpolation FFT Reconstruction

Abstract: This paper presents a windowed interpolation FFT-based reconstruction algorithm for the Hilbert transform to measure the reactive power.The method,by inverse and discrete Fourier,can accurately shift all the harmonics by 90 °.And by using the FFT algorithm of Jia Haiming window,it can calculate each harmonic frequency,phase,and amplitude calculation.It overcomes the impact of spectral leakage and eliminates its measurement error.Simulation results show that the method has high measurement accuracy.

On the computation of discrete Fourier transform using Fermat number transform

Abstract: Wan-Chi Siu, AP(HK), MPhil, CEng, MIERE, MemIEEE, and AG Constantinides, BSc(Eng), PhD, CEng, MIEE, SenMemIEEE Indexing terms: Signal processing, Discrete Fourier transforms, Fermat number transforms Abstract: In the paper the results of a study using Fermat number transforms (FNTs) to compute discrete Fourier transforms (DFTs) are presented Eight basic FNT modules are suggested and used as the basic sequence lengths to compute long DFTs The number of multiplications per point is for most cases not more than one, whereas the number of shift-adds is approximately equal to the number of additions in the Winograd- Fourier-transform algorithm and the polynomial transform Thus the present technique is very effective in computing discrete Fourier transforms tion from AN 2 to IN log2 N, while the number of real additions is reduced from 2N 2 to 3JV log2 N for N-point DFTs The major disadvantages of the FFT algorithm are that (i) it still requires quite a large number of multiplica- tions and (ii) the number of real multiplications per point is almost the same for both real and complex input data This last point makes the FFT very inefficient for the purpose of calculating the DFT of real input data In 1975, Winograd (3) showed that the minimum number of multi- plications required to compute the circular convolution of two length-JV sequences is 2N — K, where K is the number of divisors of N including 1 and N Agarwal and Cooley (4), Winograd (5) and Kolba and Parks (6) made use of Rader's theorem (7) on DFT with prime transform length to construct their algorithms for the computation of DFT Compared with conventional FFT methods, the Winograd-Fourier-transform algorithms require only one- half to one-third of the number of multiplications needed for conventional FFT, with a slightly larger number of additions per point In 1977, Nussbaumer (8) first sug- gested the use of polynomial transform over the field of polynomials for computing two-dimensional convolutions This approach led to the development of fast algorithms (9-11) for computing DFTs using polynomial transforms The method requires two to four real multiplications per point for complex input data For the computation of real input data, both the Winograd-Fourier-transform algo- rithm and polynomial transform require in fact only-half of the operations needed for the case of complex data Reed and Truong (12, 13) proposed a technique for the computation of discrete Fourier transforms, based on Win- ograd's method in combination with Mersenne prime number theoretic transforms This hybrid algorithm requires fewer multiplications than either the standard FFT or Winograd's more conventional algorithm

Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling

Abstract: Numerical simulation and inversion imaging are essential in geophysics exploration. Fourier transform plays a vital role in geophysical numerical simulation and inversion imaging, especially in solving partial differential equations. This paper proposes an arbitrary sampling Fourier transform algorithm (AS-FT) based on quadratic interpolation of shape function. Its core idea is to discretize the Fourier transform integral into the sum of finite element integrals. The quadratic shape function represents the function change in each element, and then all element integrals are calculated and accumulated. In this way, the semi-analytical solution of the Fourier oscillation operator in each integral interval can be obtained, and the Fourier transform coefficient can be calculated in advance, so the algorithm has high calculation accuracy and efficiency. Based on the one-dimensional (1D) transform, the two-dimensional (2D) transform is realized by integrating the 1D Fourier transform twice, and the three-dimensional (3D) transform is realized by integrating the 1D Fourier transform three times. The algorithm can sample flexibly according to the distribution of integrated values. The correctness and efficiency of the algorithm are verified by Fourier transform pairs. The AS-FT algorithm is applied to the numerical simulation of magnetic anomalies. The accuracy and efficiency are compared with the standard Fast Fourier transform (standard-FFT) and Gauss Fast Fourier transform (Gauss-FFT). It shows that the AS-FT algorithm has no edge effects and has a higher computational speed. The AS-FT algorithm has good adaptability to continuous medium, weak magnetic catastrophe medium, and strong magnetic catastrophe medium. It can achieve the same as or even higher accuracy than Gauss-FFT through fewer sampling points. The AS-FT algorithm provides a new means for partial differential equation solution in geophysics. It successfully solves the boundary problems, which makes it an efficient and high-precision Fourier transform approach with promising applications. Therefore, the AS-FT algorithm has excellent advantages in solving partial differential equations, providing a new means for solving geophysical forward and inverse problems.