Showing papers on "Fast Fourier transform 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
01 Sep 1976
TL;DR: The Karhunter-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, which could be useful in data compression and other mean-square signal processing applications.
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.
211 citations
TL;DR: In this paper, an alternative form of the fast Fourier transform (FFT) is developed, which has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary.
Abstract: An alternative form of the fast Fourier transform (FFT) is developed. The new algorithm has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary. The advantages of the new form would, therefore, seem to be most pronounced in systems for which multiplication are most costly.
161 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: The hardware design and implementation of aFermat number transform (FNT) is described, the arithmetic logic design is treated in detail and a new data representation for integers modulo a Fermat number is derived.
Abstract: The hardware design and implementation of a Fermat number transform (FNT) is described. The arithmetic logic design is treated in detail and a new data representation for integers modulo a Fermat number is derived. In addition, the FNT is compared with the fast Fourier transform (FFT) on the basis of hardware required for a pipeline convolver.
78 citations
10 Aug 1976
TL;DR: A Hybrid Mixed Basis FFT multiplication algorithm which has a cross-over point at degree 25 and is generally faster than a basic FFT algorithm, while retaining the desirable O(N log N) timing function of an FFT approach.
Abstract: The “fast” polynomial multiplication algorithms for dense univariate polynomials are those which are asymptotically faster than the classical O(N2) method. These “fast” algorithms suffer from a common defect that the size of the problem at which they start to be better than the classical method is quite large; so large, in fact that it is impractical to use them in an algebraic manipulation system.A number of techniques are discussed here for improving these fast algorithms. The combination of the best of these improvements results in a Hybrid Mixed Basis FFT multiplication algorithm which has a cross-over point at degree 25 and is generally faster than a basic FFT algorithm, while retaining the desirable O(N log N) timing function of an FFT approach.The application of these methods to multivariate polynomials is also discussed. The use is advocated of the Kronecker Trick to speed up a fast algorithm. This results in a method which has a cross-over point at degree 5 for bivariate polynomials. Both theoretical and empirical computing times are presented for all algorithms discussed.
77 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
Patent•
08 Nov 1976TL;DR: In this paper, a binary pseudo-random sequence generated by a shift register with feedback is used as a noise signal for testing an electrical or mechanical system, and the output is sampled and the power spectra computed by discrete Fast Fourier Transform techniques.
Abstract: A binary pseudo-random sequence generated by a shift register with feedback is used as a noise signal for testing an electrical or mechanical system, and the output is sampled and the power spectra computed by discrete Fast Fourier Transform techniques. The sequence bit interval and sampling interval have a predetermined ratio, and either the noise signal or output test signal is low pass filtered, with the result that closely spaced equal amplitude sine waves are effectively applied to the system under test and their responses separated at the output with no statistical uncertainty.
76 citations
TL;DR: An array may be reordered according to a common permutation of the digits of each of its element indices which results from common fast Fourier transform algorithms.
Abstract: An array may be reordered according to a common permutation of the digits of each of its element indices. The digit-reversed reordering which results from common fast Fourier transform (FFT) algorithms is an example. By examination of this class of permutation in detail, very efficient algorithms for transforming very long arrays are developed.
TL;DR: Two new design methods for infinite-duration impulse-response (IIR) multirate digital filters are introduced, and a state-variable implementation is given.
Abstract: Two new design methods for infinite-duration impulse-response (IIR) multirate digital filters are introduced, and a state-variable implementation is given. One method is an extension of the impulse-invariant method and results in the direct synthesis of a multi-rate filter which realizes in discrete time the response of a continuous time circuit. The second method employs linear programming (LP) to optimize the frequency domain response. For finite-length impulse response (FIR) multirate filters an efficient fast Fourier transform (FFT) implementation is presented.
IBM1
TL;DR: It is shown that under certain conditions, these transforms can be computed by means of fast transform algorithms and permit the evaluation of digital convolutions with better efficiency and accuracy than does the Fast Fourier Transform.
Abstract: Complex Mersenne Transforms are defined in a ring of integers modulo a Mersenne or pseudo-Mersenne number and can be computed without multiplications. It is shown that under certain conditions, these transforms can be computed by means of fast transform algorithms and permit the evaluation of digital convolutions with better efficiency and accuracy than does the Fast Fourier Transform.
TL;DR: It is proposed that the bandwidth be constrained when shaping the image spectrum to reduce dynamic range and introduce aliasing error in the reconstructed image.
Abstract: The use of the discrete Fourier transform in digital holography introduces aliasing error in the reconstructed image. Spectrum shaping to reduce dynamic range may also result in a serious increase in aliasing error. The effect of aliasing in digital holography is analyzed. It is proposed that the bandwidth be constrained when shaping the image spectrum. Experimental results show the approach to be quite effective.
TL;DR: In this paper, the authors describe the practical implementation of the Wiener functional expansion on a medium sized digital computer for nonlinear systems whose outputs are continuous or pulsatile signals, which is well suited to the analysis of nonlinear biological systems, particularly those encountered in neurophysiology, because of its generality, ability to deal with hard nonlinearities and ease of use with systems having pulsatile outputs.
Abstract: Nonlinear systems which have finite memories and are time invariant can be completely described by the Wiener functional expansion, in which a series of multidimensional kernels provide a polynomial approximation to the nonlinear behaviour. The kernels give a best fitting estimation to the total system behaviour in the least mean square sense and can therefore be used to describe systems in which the nonlinearities include discontinuous functions. A modification of the Wiener method described by Lee and Schetzen, which uses kernels defined in terms of cross correlation functions, has been used in most practical attempts to analyse nonlinear systems, but we have previously described how the cross correlations may be replaced with complex multiplications in the frequency domain. The speed of domain translation offered by the fast Fourier transform makes this method more efficient than time domain estimation. In this paper the practical implementation of the technique on a medium sized digital computer is described for nonlinear systems whose outputs are continuous or pulsatile signals. This description should be adequate to allow others to implement the analysis scheme. The technique is well suited to the analysis of nonlinear biological systems, particularly those encountered in neurophysiology, because of its generality, ability to deal with hard nonlinearities and ease of use with systems having pulsatile outputs.
TL;DR: One-dimensional and two-dimensional generalized discrete Fourier transforms (GFTs) are introduced in this article, and the result holds also for the DFT, as it is a particular case of the GFT.
Abstract: One-dimensional and two-dimensional generalized discrete Fourier transforms (GFT) are introduced. If a one-dimensional vector A is fractured into a two-dimensional matrix B, a one-dimensional GFT on A and a two-dimensional GFT on B give the same result and require the same number of operations to be computed. The result holds also for the DFT, as it is a particular case of the GFT.
TL;DR: Several examples are given illustrating how the Cooley–Tukey fast Fourier transform is useful as a teaching tool to introduce the subtleties of spectral analysis of sampled data by interactive minicomputer experiments.
Abstract: The Cooley–Tukey fast Fourier transform (FFT) has had an extraordinary impact on the computation of Fourier transforms. A tutorial account is given of how the algorithm works and of its relationship to the more familiar continuous Fourier transform and Fourier series. Some pitfalls associated with sampled data over a finite window are outlined. Several examples are given illustrating how the FFT is useful as a teaching tool to introduce the subtleties of spectral analysis of sampled data by interactive minicomputer experiments. A bibliography to the literature is given.
01 Oct 1976
TL;DR: In this article, a digital processing algorithm and its associated system design for producing images from Synthetic Aperture Radar (SAR) data is described. But the proposed system uses the Fast Fourier Transform (FFT) approach to perform the two-dimensional correlation process.
Abstract: This paper describes a digital processing algorithm and its associated system design for producing images from Synthetic Aperture Radar (SAR) data. The proposed system uses the Fast Fourier Transform (FFT) approach to perform the two-dimensional correlation process. The range migration problem, which is often a major obstacle to efficient processing, can be alleviated by approximating the locus of echoes from a point target by several linear segments. SAR data corresponding to each segment is correlated separately, and the results are coherently summed to produce full-resolution images. This processing approach exhibits greatly improved computation efficiency relative to conventional digital processing methods.
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.
Patent•
27 Dec 1976
TL;DR: In this article, an incremental digital filter provides high speed and low cost capability such as for performing fast Fourier transforms (FFTs), correlations, convolutions, and other digital filter operations.
Abstract: An incremental digital filter provides high speed and low cost capability such as for performing fast Fourier transforms (FFTs), correlations, convolutions, and other digital filter operations. One configuration operates at microwave sample rates, computing a complete 512-point FFT in 0.2 microseconds for an effective sample rate of 2.56 gigahertz. High speed and low cost are derived from a parallel pipeline architecture in combination with incremental processing. Parallel pipeline architecture provides extremely high speed while the incremental mechanization provides a simple arrangement with a low component count for low cost. The incremental nature of the processor provides an integrating type mechanization, where integration-after-transformation yields high processing gain for signal-to-noise-ratio enhancement. High data rate input and output mechanizations are provided to accommodate the high processing rates. An improved input mechanization involves analog signal to incremental digital conversion. An improved output mechanization involves integration after filtering for data rate reductions and for signal enhancement and also involves a bus output structure for multiplexing of output parameters.
TL;DR: A technique for processing FFT outputs to realize banks of narrowband filters for which spectral band centers and spectral bandwidths may be arbitrarily assigned is developed.
TL;DR: Criteria for choosing an algorithm based on the data type and specific methods for both of these approaches are developed with comparisons of accuracy and efficiency are given.
TL;DR: A fast sequency ordered Walsh transform was proposed in this paper, which accepts data in normal order, returning the coefficients in bit-reversed sequency order, and is the complement to one developed by Manz.
Abstract: A fast sequency ordered Walsh Transform algorithm is presented, which is the complement to one developed by Manz. It is in place, is its own inverse, and accepts data in normal order, returning the coefficients in bit-reversed sequency order.
TL;DR: In this paper, a digital memory device and a microcomputer are introduced to calculate correlation functions and Fourier transform in voltammetric measurements, and typical cross-spectra for simple diffusion and kinetic processes are obtained.
TL;DR: Tracing the factorization property provides a unity to the basic concepts involved in the development of cyclic reduction, predictor-corrector, splitting, fast Fourier transform, and pseudospectrai methods.
Abstract: Recent developments in the use of numerical methods for fluid flow simulation show an increasing tendency to use numerical operators that can, in various ways, be factored. Use of methods having this property often increases the accuracy and efficiency of computer codes. Tracing the factorization property provides a unity to the basic concepts involved in the development of cyclic reduction, predictor-corrector, splitting, fast Fourier transform, and pseudospectral methods.-
TL;DR: In this article, a special number-theoretic transform that can be computed using a high-radix fast Fourier transform is defined on primes of the form (2n? 1) 2n + 1.
Abstract: A special number-theoretic transform that can be computed, using a high-radix fast Fourier transform, is defined on primes of the form (2n ? 1) 2n +1. Methods for finding these primes and the primitive dth roots of unity in a field modulo such primes are also included.
TL;DR: Using the fast Fourier transform (FFT) algorithm, computer-reconstructed images of radio sources are obtained from conventional height gain measurements over line-of-sight propagation paths for microwave frequencies of 2 and 4 GHz.
Abstract: Using the fast Fourier transform (FFT) algorithm, computer-reconstructed images of radio sources are obtained from conventional height-gain measurements over line-of-sight propagation paths for microwave frequencies of 2 and 4 GHz. The height-gain curves are regarded as one-dimensional microwave holograms.