Showing papers on "Split-radix FFT algorithm published in 1976"
TL;DR: In this paper, the Karhunen-Loeve transform for a class of signals is proven to be a set of periodic sine functions and this k-means expansion can be obtained via an FFT algorithm.
Abstract: The Karhunen-Loeve transform for a class of signals is proven to be a set of periodic sine functions and this Karhunen-Loeve series expansion can be obtained via an FFT algorithm. This fast algorithm obtained could be useful in data compression and other mean-square signal processing applications.
215 citations
TL;DR: Significant time-saving can be achieved by a simple modification to the radix-2 decimation in-time fast Fourier transform (FFT) algorithm when the data sequence to be transformed contains a large number of zero-valued samples.
Abstract: Significant time-saving can be achieved by a simple modification to the radix-2 decimation in-time fast Fourier transform (FFT) algorithm when the data sequence to be transformed contains a large number of zero-valued samples. The time-saving is accomplished by replacing M - L stages of the FFT computation with a simple recopying procedure where 2Mis the total number of points to be transformed of which only 2Lare nonzero.
138 citations
TL;DR: A particularly simple way to control fast Fourier transform (FFT) hardware that allows parallel organization of the memory such that at any stage the two inputs and outputs of each butterfly belong to different memory units, hence can always be accessed in parallel.
Abstract: A particularly simple way to control fast Fourier transform (FFT) hardware is described. The method produces the indices both for inputs of each butterfly operation and for the appropriate W. In addition, this method allows parallel organization of the memory such that at any stage the two inputs and outputs of each butterfly belong to different memory units, hence can always be accessed in parallel.
108 citations
TL;DR: This paper deals with two's complement arithmetic with either rounding or chopping with eitherRoundoff errors for radix-2 FFT's and mixed-radix FFTs.
Abstract: A statistical model for roundoff errors is used to predict the output noise of the two common forms of the fast Fourier transform (FFT) algorithm, the decimations in-time and in-frequency. This paper deals with two's complement arithmetic with either rounding or chopping. The total mean-square errors and the mean-square errors for the individual points are derived for radix-2 FFT's. Results for mixed-radix FFT are also given.
93 citations
TL;DR: This correspondence shows that the amount of work can be cut to doing two single length FFT's, which is equivalent to doing one double length fast Fourier transform.
Abstract: Ahmed has shown that a discrete cosine transform can be implemented by doing one double length fast Fourier transform (FFT). In this correspondence, we show that the amount of work can be cut to doing two single length FFT's.
77 citations
TL;DR: An algorithm to find the coefficients of the s -polynomial D (s) = |H(s)| is obtained, where H(s) is an arbitrary s - polynomial square matrix.
Abstract: An algorithm to find the coefficients of the s -polynomial D(s) = |H(s)| is obtained, where H(s) is an arbitrary s -polynomial square matrix. The algorithm, based on the fast Fourier transform (FFT), is of an order of magnitude faster than existing methods.
32 citations
TL;DR: This chapter discusses application of fast Fourier transform (FFT) in radio astronomy and it is shown how this algorithm is programmed on a digital computer.
Abstract: Publisher Summary This chapter discusses application of fast Fourier transform (FFT) in radio astronomy. The Fourier transform is a particularly useful computational technique in radio astronomy. The essence of the FFT technique is that it is possible to treat the one-dimensional DFT as though it were a pseudo-two-dimensional one, and then reduce the running time by performing the inner and outer summations separately. The basic idea behind the FFT is discussed and it is shown how this algorithm is programmed on a digital computer. Because of the requirement for computational speed, a number of programs are given. These include short, moderately efficient subroutines for the transform of one-dimensional, complex data (FOURG and FOURI). With the addition of a subroutine (FXRLI) to either of the above routines, real, one-dimensional data may be transformed in half the time with half the memory storage. Additional subroutines (CFFT2, RFFT2, and HFFT2) permit the transform of two-dimensional data. A program is also given for transforming real, symmetric data for which only the cosine (or sine) transform is desired (FORSI).
12 citations
20 Oct 1976
TL;DR: The abstract character of the FFT, in particular its role as an algebraic algorithm, is what this paper is about.
Abstract: In the past decade the Cooley-Tukey fast Fourier transform (FFT) [1] has achieved the status of a “super” algorithm. As a numerical (complex field) algorithm, the FFT has revolutionized large scale time series analysis in a way that counts most—economic. (See, e.g., Refs. 3-6.) Since the late sixties, the FFT has also emerged as an important algebraic(abstract field) algorithm, with many interesting applications to the theory and practice of algebraic computing. The abstract character of the FFT, in particular its role as an algebraic algorithm, is what this paper is about.Our discussion centres around the following questions:1. What is the discreteFourier transform?2. What is the fastFourier transform?3. What is its role in algebraiccomputing?4. Is a finite field(mod p) FFT feasible?
10 citations
TL;DR: The nature of the signal has been exploited to reduce to a minimum the number of multiplications and the calculations are performed in an ordered sequence in order to evaluate only the nonredundant terms at each pass.
Abstract: This article describes the implementation of a modular fast Fourier transform (FFT) processor for real-input applications. The nature of the signal has been exploited to reduce to a minimum the number of multiplications and the calculations are performed in an ordered sequence in order to evaluate only the nonredundant terms at each pass. The number of components required for transforming N points is given as a unction of the number of passes. A processing rate of one point per clock cycle at frequencies up to 10 MHz is realizable making the processor ideally suited for a number of real time computations.
6 citations
TL;DR: In this article, it was shown that the FFT method is equivalent to the lagged product method in terms of spectral window, and that the only difference between them lies in regard to the spectral window.
Abstract: Two conventional methods of computing the power spectrum, via the autocovariance function or via the fast Fourier transform (referred to as the lagged product method and the FFT method respectively for simplicity), have been examined analytically and numerically for equally spaced time series of finite length. It is found that the two methods are equivalent to each other, and that the only difference between them lies in regard to the spectral window. Spectral windows for the FFT method are superior to those for the lagged product method in that they do not show any negative values and that their influence is band-limited in frequency domain. There is little difference in spectral estimates between the two methods. In many cases the FFT method is economical in computation time, but for the case of large data points and small maximum lag the lagged product method is the more economical. It is proved that in the strict sense the power spectrum for higher frequencies than the Nyquist frequency is no...
5 citations
TL;DR: This work derives a simple and easily applied upper bound on the increase in the distortion-rate function (DRF) for the mean-squared error criterion incurred by substitution of the DFT for the KLT.
Abstract: The discrete Fourier transform (DFT) often is used instead of the optimum Karhunen-Lo \grave{e} ve transform (KLT) in encoding a stationary normal time series, because the recursive FFT is computationally efficient and yields "nearly" uncorrelated components. Substituting the DFT for the KLT and then treating its components as if they were uncorrelated reduces the ultimate performance attainable in fixed-rate source coding. We address the problem of this performance degradation by deriving a simple and easily applied upper bound on the increase in the distortion-rate function (DRF) for the mean-squared error criterion incurred by substitution of the DFT for the KLT.