Showing papers on "Discrete sine transform published in 1976"
TL;DR: A method is presented for numerically inverting a Laplace transform that requires, in addition to the transform function itself, only sine, cosine, and exponential functions and includes a transformation of the approximating series into one that converges very rapidly.
Abstract: A method is presented for numerically inverting a Laplace transform that requires, in addition to the transform function itself, only sine, cosine, and exponential functions. The method is conceptually much like the method of Dubner and Abate, which approximates the inverse function by means of a Fourier cosine series. The method presented here, however, differs from theirs in two important respects. First of all, the Fourier series contains additional terms involving the sine function selected such that the error in the approximation is less than that of Dubner and Abate and such that the Fourier series approximates the inverse function on an interval of twice the length of the corresponding interval in Dubner and Abate's method. Second, there is incorporated into the method in this paper a transformation of the approximating series into one that converges very rapidly. In test problems using the method it has routinely been possible to evaluate inverse transforms with considerable accuracy over a wide range of values of the independent variable using a relatively few determinations of the Laplace transform itself.
697 citations
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: In this article, the authors compared the effectiveness of the discrete cosine and Fourier transforms in decorrelating sampled signals with Markov-1 statistics, and showed that the DCT offers a higher (or equal) effectiveness than the discrete Fourier transform for all values of the correlation coefficient.
Abstract: This correspondence compares the effectiveness of the discrete cosine and Fourier transforms in decorrelating sampled signals with Markov-1 statistics. It is shown that the discrete cosine transform (DCT) offers a higher (or equal) effectiveness than the discrete Fourier transform (DFT) for all values of the correlation coefficient. The mean residual correlation is shown to vanish as the inverse square root of the sample size.
116 citations
TL;DR: The scale invariance of the Mellin transform and its optical synthesis in real-time are discussed in this article, where a scale invariant correlation and a combined Fourier-Mellin transform that is both scale and shift invariant are discussed.
Abstract: The scale invariance of the Mellin transform and its optical synthesis in real-time are discussed. An initial off-line demonstration of an optical Mellin transform is presented. A scale invariant correlation and a combined Fourier-Mellin transform that is both scale and shift invariant are discussed. Applications of these transforms in optical data processing and optical pattern recognition are emphasized.
87 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: 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.
47 citations
01 Mar 1976
TL;DR: In this paper, an efficient algorithm for the DFT of N point symmetric real-valued series with time samples taken as odd multiples of half the sampling period T/2 and frequency samples are taken as 1/2NT is presented.
Abstract: An efficient algorithm is obtained for the DFT of N point symmetric real-valued series if time samples are taken as odd multiples of half the sampling period T/2 and frequency samples are taken as odd multiples of 1/2NT.
41 citations
34 citations
24 citations
TL;DR: The problem of evaluating successively the discrete Fourier transform on ordered sets of N elements staggered of M is considered, and three procedures for solving such a problem are given, of which two are recursive and one nonrecursive.
Abstract: In this work the problem of evaluating successively the discrete Fourier transform (DFT) on ordered sets of N elements staggered of M is considered. Three procedures for solving such a problem are given, of which two are recursive and one nonrecursive. The complexity of each procedure, in number of complex multiplications, is about (N/2) \log_{2} 4M .
15 citations
Patent•
01 Mar 1976
TL;DR: In this article, an arrangement for computing the discrete Fourier transform intended for converting N samples of a real signal in the time domain to N real Fourier coefficients is presented. But this device is implemented with a conventional Fourier transformer of the order N/4, to which an input computer unit and an output computer unit are connected in which a small number of multiplications of complex numbers is performed.
Abstract: An arrangement for computing the discrete Fourier transform intended for converting N samples of a real signal in the time domain to N real Fourier coefficients. This device is implemented with a conventional Fourier transformer of the order N/4, to which an input computer unit and an output computer unit are connected in which a small number of multiplications of complex numbers is performed.
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).
TL;DR: The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept.
Abstract: Truncated series expansion is used to obtain discrete-time windows which are optimal in the least-square sense for a given number of terms. The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept. As a consequence, the resulting windows are easy to obtain and economical to implement in practical applications.
12 Apr 1976
TL;DR: It is shown that the Discrete Fourier Transform, when used in the conventional manner with the frequency samples located at zero and integer multiples of 1/T, gives an inaccurate representation of the spectrum of certain frequencies that are located near the top and bottom end of the band.
Abstract: It is shown that the Discrete Fourier Transform (DFT), when used in the conventional manner with the frequency samples located at zero and integer multiples of 1/T, where T is the signal duration, gives an inaccurate representation of the spectrum of certain frequencies that are located near the top and bottom end of the band. It is further shown that this type of error can be eliminated by using the Odd Discrete Fourier Transform (ODFT) in which the frequency samples are located at odd multiples of 1/2T. An application of the ODFT in two dimensional filtering is also discussed.
01 Sep 1976
TL;DR: In this paper, a matrix formulation of the discrete Hilbert transform is presented, which has the advantage of reducing the number of multiplications by a factor of two, compared to the matrix formulation given by Burris.
Abstract: A matrix formulation of the discrete Hilbert transform, an alternative to that given by Burris [1], is presented. This has the advantage of reducing the number of multiplications by a factor of two.
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.
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.
01 May 1976
TL;DR: Starting from the definition of the Fourier transform, the matrix expressions of both one- and two-dimensional Fourier transforms and their inverse transforms are given, which show that the discrete Fourier transformations are closely related to the nature of a balanced polyphase system.
Abstract: Starting from the definition of the Fourier transform, the matrix expressions of both one- and two-dimensional Fourier transforms and their inverse transforms are given, which show that the discrete Fourier transforms and their inverse transforms are closely related to the nature of a balanced polyphase system. The system structures are derived from the matrix expressions, which are shown diagramatically. A special feature of the system structure is that the main devices necessary for hardware implementation are tapped delay lines, product operators, adders, and a sinusoidal wave generator.
TL;DR: In this work three structures are presented for evaluating in real time the discrete Fourier transform on successive sections of a sampled signal.
Abstract: In this work three structures are presented for evaluating in real time the discrete Fourier transform (DFT) on successive sections of a sampled signal. Each section consists of N elements and two successive sections are staggered of M \leq N elements. Each structure evaluates N/M DFT coefficients at every step and requires about (N/(2M)) \log_{2} 4M multipliers.
01 Apr 1976
TL;DR: It is demonstrated that the double-sampling technique in digital signal processing produces the effects similar to the beat-down in a heterodyne system of analog circuits.
Abstract: An effective method for calculating the Discrete Fourier Transform (DFT) of a double-sampled sequence (a pair of the equi-interval-sampled sequences) is presented in a matrix form. The calculation formula shows that the DFT of a certain type of band-limited signals can be effectively evaluated from the DFT's of a pair of the equi-interval-sampled sequences of the signal. It is demonstrated that the double-sampling technique in digital signal processing produces the effects similar to the beat-down in a heterodyne system of analog circuits. A calculation example is given in detail and some computational and practical applications are described briefly.
TL;DR: It is shown that the two- dimensional processing performed according to two significant two-dimensional decomposition rules involves exactly the same operations on the same data as the one-dimensional processing.
Abstract: Two significant two-dimensional decomposition rules for the Discrete Fourier Transform of a set ofN data (N=2p) are considered. It is shown that the two-dimensional processing performed according to such rules involves exactly the same operations on the same data as the one-dimensional processing. This means that, if the same rule is iteratively applied with arbitrary dimensions, always the same fast algorithm is obtained.