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Discrete sine transform
About: Discrete sine transform is a research topic. Over the lifetime, 3269 publications have been published within this topic receiving 73181 citations.
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TL;DR: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable and promises to speed up Fourier-transform calculations.
Abstract: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ HartleyR. V. L., Proc. IRE30, 144 ( 1942)] as the DFT bears to the Fourier transform.
465 citations
TL;DR: In this paper, a new scheme is presented for the determination of the parameters that characterize a multifrequency signal, where the signal is weighted before the discrete Fourier transform (DFT) is calculated from which the frequencies and complex amplitudes of the various components of the signal are obtained by interpolation.
Abstract: A new scheme is presented for the determination of the parameters that characterize a multifrequency signal. The essential innovation is that the signal is weighted before the discrete Fourier transform (DFT) is calculated from which the frequencies and complex amplitudes of the various components of the signal are obtained by interpolation. It is shown that by using the Hanning window for tapering substantial improvements are achieved in the following respects: i) more accurate results are obtained for interpolated frequencies, etc., ii) harmonic interference is much less troublesome even if many tones with comparable strengths are present in the spectrum, iii) nonperiodic signals can be handled without an a priori knowledge of the tone frequencies. The stability of the new method with respect to noise and arithmetic roundoff errors is carefully examined.
440 citations
TL;DR: A transform analogous to the discrete Fourier transform may be defined in a finite field, and may be calculated efficiently by the Fast Fourier Transform (FFT) algorithm as discussed by the authors.
Abstract: A transform analogous to the discrete Fourier transform may be defined in a finite field, and may be calculated efficiently by the 'fast Fourier transform' algorithm. The transform may be applied to the problem of calculating convolutions of long integer sequences by means of integer arithmetic.
431 citations
TL;DR: It is shown that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves.
Abstract: This paper describes the discrete Radon transform (DRT) and the exact inversion algorithm for it. Similar to the discrete Fourier transform (DFT), the DRT is defined for periodic vector-sequences and studied as a transform in its own right. Casting the forward transform as a matrix-vector multiplication, the key observation is that the matrix-although very large-has a block-circulant structure. This observation allows construction of fast direct and inverse transforms. Moreover, we show that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves. In fact, we describe not a single transform, but a class of transforms, representatives of which correspond in one way or another to discrete versions of the RT and its generalizations. An interesting observation is that the exact inversion algorithm cannot be obtained directly from Radon's inversion formula. Given the fact that the RT has no nontrivial one-dimensional analog, exact invertibility makes the DRT a useful tool geared specifically for multidimensional digital signal processing. Exact invertibility of the DRT, flexibility in its definition, and fast computational algorithm affect present applications and open possibilities for new ones. Some of these applications are discussed in the paper.
426 citations
01 Jan 2007
TL;DR: In this article, a calculus of fractional finite differences is introduced and a discrete Laplace transform method is developed to obtain solutions to a family of finite fractional difference equations and employ the transform method to obtain the solutions.
Abstract: We begin with an introduction to a calculus of fractional finite differences. We extend the discrete Laplace transform to develop a discrete transform method. We define a family of finite fractional difference equations and employ the transform method to obtain solutions. AMS subject classification: 39A12, 26A33.
425 citations