Showing papers on "Discrete Hartley transform published in 1980"
Book•
01 Jan 1980
TL;DR: In this paper, Fourier transform is used for spectral analysis of periodical signals and some properties of the spectrum are analyzed, and it is demonstrated that the spectrum is strongly depended of signal duration.
Abstract: This paper analyses Fourier transform used for spectral analysis of periodical signals and emphasizes some of its properties. It is demonstrated that the spectrum is strongly depended of signal duration that is very important for very short signals which have a very rich spectrum, even for totally harmonic signals. Surprisingly is taken the conclusion that spectral function of harmonic signals with infinite duration is identically with Dirac function and more of this no matter of duration, it respects Heisenberg fourth uncertainty equation. In comparison with Fourier series, the spectrum which is emphasized by Fourier transform doesn’t have maximum amplitudes for signals frequencies but only if the signal lasting a lot of time, in the other situations these maximum values are strongly de-phased while the signal time decreasing. That is why one can consider that Fourier series is useful especially for interpolation of nonharmonic periodical functions using harmonic functions and less for spectral analysis. Key-Words — signals, Fourier transform, continuous spectrum properties, Quantum Physics, Fourier series, discrete spectrum
609 citations
TL;DR: The discrete cosine transform (DCT) of an N-point real signal is derived by taking the discrete Fourier transform (DFT) of a 2N-point even extension of the signal and the method is extended to two dimensions, with a saving of 1/4 over the traditional method that uses the DFT.
Abstract: The discrete cosine transform (DCT) of an N-point real signal is derived by taking the discrete Fourier transform (DFT) of a 2N-point even extension of the signal. It is shown that the same result may be obtained using only an N-point DFT of a reordered version of the original signal, with a resulting saving of 1/2. If the fast Fourier transform (FFT) is used to compute the DFT, the result is a fast cosine transform (FCT) that can be computed using on the order of N \log_{2} N real multiplications. The method is then extended to two dimensions, with a saving of 1/4 over the traditional method that uses the DFT.
334 citations
TL;DR: A sparse-matrix factorization is developed for the discrete sine transform (DST) that has a recursive structure and leads directly to an efficient algorithm for implementing the DST, a feature most desirable and very similar ot that of the DCT.
Abstract: A sparse-matrix factorization is developed for the discrete sine transform (DST). This factorization has a recursive structure and leads directly to an efficient algorithm for implementing the DST, a feature most desirable and very similar ot that of the DCT. This algorithm requires fewer arithmetic operations compared to that for the discrete cosine transform (DCT).
87 citations
TL;DR: In this article, the effectiveness of the discrete sine transform (DST) in terms of residual correlation as developed by Hamidi and Pearl[1] is investigated. And the odd-even property of the DST is established.
30 citations
TL;DR: Analysis is given for Good's algorithm and for two algorithms that compute the discrete Fourier transform in O(n log n) operations: the chirp-z transform and the mixed-radix algorithm that computes the transform of a series of prime length p in P log p operations.
25 citations
TL;DR: In this paper, conditions for a transform of the DFT structure, defined in a ring of residues of algebraic integers, to map cyclic convolution isomorphically into a pointwise product are presented.
Abstract: Conditions are presented for a transform of the DFT structure, defined in a ring of residues of a ring of algebraic integers, to map cyclic convolution isomorphically into a pointwise product. The conditions are used to verify that a number of potentially useful transforms (which require no general multiplications) satisfy this property. In particular, transforms defined in residue rings of the Gaussian integers, the Eisenstein integers, and a biquadratic domain are studied.
20 citations
Patent•
IBM1
TL;DR: In this article, an apparatus for computing the two-dimensional discrete Fourier transform (DFT) of an image comprised of N×N samples is presented. But it is not shown how to compute the one-dimensional DFT of the image.
Abstract: An apparatus for computing the two-dimensional discrete Fourier transform (DFT) of an image comprised of N×N samples. The samples within each row are respectively multiplied by W-n.sbsp.1, n1 =0, 1, . . . , N-1 and stored in a memory 17. A device 20 derives therefrom N polynomials of N terms by means of a polynomial transform. The terms of each of these polynomials are multiplied by Wn.sbsp.1 and a device 28 computes the one-dimensional DFT thereof, thereby providing the N2 terms of the transform of said image.
14 citations
TL;DR: By introducing a number of zeros as the elements in the transform matrices, a modified version of the transforms can be computed efficiently and can be used in the general area of information processing.
4 citations
TL;DR: Three applications of a theorem of I. J. Good are made: a simplification of the fast Hadamard transform algorithm of R.R. Green used in the decoding of first-order Reed-Muller codes, a simple proof of the well-known Parseval formula, and an application to the discrete one-dimensional Fourier transform.
Abstract: Three applications of a theorem of I. J. Good are made: a simplification of the fast Hadamard transform algorithm of R.R. Green used in the decoding of first-order Reed-Muller codes, a simple proof of the well-known Parseval formula, and an application to the discrete one-dimensional Fourier transform.
4 citations
IBM1
TL;DR: A new fast computation algorithm for multidimensional DFTs by using a single polynomial transform and auxiliary calculations and all calculations are performed with a reduced number of additions by using FFT-type algorithms.
Abstract: In this paper, we introduce a new fast computation algorithm for multidimensional DFTs. This method maps efficiently some multidimensional DFTs into one-dimensional DFTs by using a single polynomial transform and auxiliary calculations. The polynomial transform is computed without multiplications and all calculations are performed with a reduced number of additions by using FFT-type algorithms. The relationship with earlier polynomial transform approaches is explored and it is shown that the new method yields a simpler structure at the expense of a slight increase in number of arithmetic operations. Various techniques for reducing the auxiliary calculations are investigated and schemes which combine different polynomial transform techniques are presented.
4 citations
09 Apr 1980
TL;DR: A procedure is described for the computation of the discrete cosine transform (DCT) via the use of the arcsine transform, which eliminates time-consuming multiplications, the DCT being accomplished with only additions and table lookups.
Abstract: A procedure is described for the computation of the discrete cosine transform (DCT) via the use of the arcsine transform. The approach eliminates time-consuming multiplications, the DCT being accomplished with only additions and table lookups. While a fast Fourier transform (FFT) approach to computing the DCT involves on the order of N\log_{2}2N "butterfly" computations to evaluate all N coefficients, the arcsine method requires only 4N - 1 real additions and 2N table lookups to evaluate each DCT coefficient. Thus, for applications in which M coefficients are desired or when N is reasonably small (say, N \leq 256 ), the arcsine approach is favored over that of the FFT. Some approaches to hardware implementation are presented.
01 Apr 1980
TL;DR: An explicit form for the polar representation of the DFT is presented, powers of theDFT are studied, and possible applications to signal multiplexing and transform coding are suggested.
Abstract: We study the problem of diagonalizing the DFT by finding eigenvectors through the use of commuting operators. An explicit form for the polar representation of the DFT is presented and powers of the DFT are studied. Possible applications to signal multiplexing and transform coding are suggested.
TL;DR: An efficient method of achieving flexibility in the length of discrete convolutions, computed using Fourier and Fourier-like fast transform algorithms, is described and extended to discrete multidimensional convolutions computed using polynomial transforms.
TL;DR: Physically speaking, the edge effects become insignificant for large enough blocks, and these large blocks are entirely consistent with the aim of encoding optimally large amounts of data in a DFT.
Abstract: Codes. New York: Elsevier North-Holland, 1977. posed on the N-sequence, making the sequence u(k) wide-sense 121 R. J. Lechner, “Harmonic analysis of switching functions,” in Recent stationary with covariance relationships given in (2).’ Moreover, Developments in Switching Theoy, A. Mukhopadhyay, Ed. New York: in the limit of large N, with or without the random shift, as Academic, 1971, pp. 122-230. [31 M. A. Harrison, “Counting theorems and their applications to classificastated in the paper the eigenvalue distributions of K(p) and tion of switching functions,” in Recent Dmelopments in Switching Theory, R,(p) approach each other. That is all that is required for A. Mukhopadhyay, Ed. New York: Academic, 1971, pp. 86-122. deriving coding bounds. h,(p) is the appropriate circulant approximation to the Toeplitz form R,(p). Physically speaking, the edge effects become insignificant for large enough blocks. These large blocks are entirely consistent with the aim of encoding optimally large amounts of data in a DFT. Optimal source coding, as addressed in this paper, requires large N.
TL;DR: In this paper, a systematic technique for synthesizing and efficiently performing large discrete Fourier transformations (DFT) in the range from 60-5000 points is presented and compared with the fast Fourier transform (FFT).
Abstract: A systematic technique is presented for synthesizing and efficiently performing large discrete Fourier transformations (DFT) in the range from 60-5000 points. Computational complexity is estimated and compared with the fast Fourier transform (FFT). Prime power pairs are found which minimize the computational complexity.