Showing papers on "Split-radix FFT algorithm published in 1979"

Very Fast Fourier Transform Algorithms Hardware for Implementation

Abstract: A new method of deriving very fast Fourier transform (FFT) algorithms is described. The resulting algorithms do not employ multiplication and have a form suitable for high performance hardware implementations. The complexity of the algorithms compares favorably to the recent results of Winograd [1].

Properties of the Multidimensional Generalized Discrete Fourier Transform

Abstract: In this work the generalized discrete Fourier transform (GFT), which includes the DFT as a particular case, is considered. Two pairs of fast algorithms for evaluating a multidimensional GFT are given (T-algorithm, F-algorithm, and T′-algorithm, F′-algorithm). It is shown that in the case of the DFT of a vector, the T-algorithm represents a form of the classical FFT algorithm based on a decimation in time, and the F-algorithm represents a form of the classical FFT algorithm based on decimation in frequency. Moreover, it is shown that the T′-algorithm and the T-algorithm involve exactly the same arithmetic operations on the same data. The same property holds for the F′-algorithm and the F-algorithm. The relevance of such algorithms is discussed, and it is shown that the T′-algorithm and the F′-algorithm are particularly advantageous for evaluating the DFT of large sets of data.

Efficient FFT Simulation of Digital Time Sequences

Abstract: A stacked inverse finite Fourier transform (FFT) algorithm is presented that will efficiently synthesize a discrete random time sequence of N values from only N/2 complex values having a desired known spectral representation. This stacked inverse FFT algorithm is compatible with the synthesis of discrete random time sequences that are used with the more desirable periodic-random type of dynamic testing systems used to compute complex-valued transfer functions by the frequency-sweep method. An application to the generation of large random surface gravity waves by a hinged wavemaker in a large-scale wave flume demonstrates excellent agreement between the desired theoretical spectral representation and the smoothed, measured spectral representation for two types of two-parameter theoretical spectra as a result of the lengthier realization made possible by the stacked FFT algorithm.

On the use of symmetry in FFT computation

Abstract: It is well known that if a finite duration, N-point sequence x(n) possesses certain symmetries, the computation of its discrete Fourier transform (DFT) can be obtained from an FFT of size N/2 or smaller. This is accomplished by first preprocessing the sequence, taking the FFT of the processed sequence, and then postprocessing the results to give the desired transform. In this paper we show how a similar approach can be used for sequences which are known to have only odd harmonics. The approach is shown to be essentially the dual of the known method for time symmetry. Computer programs are included for implementing the special procedures discussed in this paper.

A New Hybrid Algorithm for Computing a Fast Discrete Fourier Transform

Abstract: In this paper for certain long transform lengths, Winograd's algorithm for computing the discrete Fourier transform (DFT) is extended considerably. This is accomplisbed by performing the cyclic convolution, required by Winograd's method, with the Mersenne prime number-theoretic transform developed originally by Rader. This new algorithm requires fewer multiplications than either the standard fast Fourier transform (FFT) or Winograd's more conventional algorithm. However, more additions are required.

Interpolation by fast Fourier and Chebyshev transforms

Abstract: Transform methods for the interpolation of regularly spaced data are described, based on fast evaluation using discrete Fourier transforms. For periodic data adequately sampled, the fast Fourier transform (FFT) is used directly. With undersampled or aperiodic data, a Chebyshev interpolating polynomial is evaluated by means of the FFT to provide minimum deviation and distributed ripple. The merits of two kinds of Chebyshev series are compared. All the methods described produce an interpolation passing directly through the given values and are applied easily to the multi-dimensional case.

Algorithm 545: An Optimized Mass Storage FFT [C6]

Abstract: The program is an implementation of the optimal sorting algorithm of the author [8] which allows a base-2 version of the Cooley-Tukey FFT algorithm [2-4] efficient access to a mass store array. Optimal sorting for the mass storage FFT has been determined independently by DeLotto and Dotti [5, 6], but in the author's version the emphasis is on \"in-place\" array modification. This results in slightly higher mass store I /O than the minimum, but requires no additional mass store working space. The method is a logical extension of the work of Singleton [9] and Brenner [1]. The program computes in place the discrete Fourier transform of a onedimensional or a multidimensional array. In the one-dimensional case the transform is defined by

The Problem of Calculating the Fast-Fourier Transform of a Step-Like Sampled Function

Abstract: The calculation of the fast fourier transform (FFT) for a step-like bounded function with unequal values at boundaries may be performed by using a convenient decomposition of the total curve into two elementary ones, one of them being a linear ramp. The method may be generalized to functions having asymptotic tails which may be approximated by simple analytic functions, the theoretical FFT of which is known.

Multidimensional fast-Fourier-transform algorithm

Abstract: A multidimensional fast-Fourier-transform algorithm is developed for the computation of multidimensional Fourier and Fourier-like discrete transforms; it has considerably less multiplications than the conventional fast-Fourier-transform methods.

A comparison of WFTA and FFT programs

Abstract: Bounds on the minimum number of data transfers (i.e., loads and stores) required by WFTA (Winograd Fourier Transform Algorithm) and FFT programs are presented. The analysis is applicable to those general-purpose computers with a small number of general processor registers (e.g., the IBM370, PDP-11, etc.). It is shown that the 1008-point WFTA requires about 21% more data transfers than the 1024-point radix-4 FFT; on the other hand, the 120-point WFTA has about 22% fewer data transfers than the 128-point radix-2 FFT. Finally, comparisons of the 'total' program execution times (multiplications, additions, and data transfers, but not indexing or permutations) are presented.

Bounds on Error Due to Finite Arithmetic in FFT Computation

Abstract: For the decimation-in-frequency FFT algorithm using fixed point arithmetic, it is shown that there are position-dependent bounds on the error amplitude in the Fourier co-efficients. This means that the error statistics are position dependent and the earlier results on finite arithmetic effects in FFT calculation are inaccurate to that extent. These results lead to worst-case deterministic design of FFT processor.

Parallelism in the computation of the FFT and the WFTA

Abstract: Arithmetic concurrencies, such as those found in special-purpose fast Fourier transform (FFT) hard-ware, are surveyed and categorized. Similar structures are then derived for the Winograd Fourier transform algorithm (WFTA). Relative time-efficiency plots are obtained for the 1024-point radix-4 FFT and the 1008-point WFTA as a function of the number of real arithmetic operations executable in parallel. This comparison shows that the relative time efficiency of the two algorithms in sequential computations generally carries over to cases where arithmetic parallelism is exploited.