Showing papers on "Discrete sine transform published in 1968"
01 Jun 1968
TL;DR: The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation, which permits the discrete Fouriers transform to be computed by means of a fast Fouriertransform algorithm, with the associated increase in speed, even though N is prime.
Abstract: The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation. This can be recognized by rearranging the members of the sequence and the transform according to a rule involving a primitive root of N. This observation permits the discrete Fourier transform to be computed by means of a fast Fourier transform algorithm, with the associated increase in speed, even though N is prime.
523 citations
TL;DR: In this article, several basic power-spectrum estimation procedures are reviewed and their statistical and mathematical properties are discussed and compared with the standard procedure that uses the cosine transform of the estimated correlation function.
Abstract: The computation of power spectra, cross spectra, coherence, and bispectra of various types of geophysical random processes is part of the established routine. Since it is routine, some of the standard procedures need to be examined rather carefully to be certain that the assumptions behind the procedures are applicable to the data on hand. The basic criteria for a particular method are its resolution bandwidth, its variance, and its bias. In this paper several basic power-spectrum estimation procedures are reviewed and their statistical and mathematical properties are discussed. The direct use of the discrete Fourier transform for various spectrum calculations is discussed in detail, and its properties are compared with the standard procedure that uses the cosine transform of the estimated correlation function.
166 citations
TL;DR: In this article, a new procedure for calculating the complex, discrete Fourier transform of real-valued time series is presented for an example where the number of points in the series is an integral power of two.
Abstract: A new procedure is presented for calculating the complex, discrete Fourier transform of real-valued time series. This procedure is described for an example where the number of points in the series is an integral power of two. This algorithm preserves the order and symmetry of the Cooley-Tukey fast Fourier transform algorithm while effecting the two-to-one reduction in computation and storage which can be achieved when the series is real. Also discussed are hardware and software implementations of the algorithm which perform only (N/4) log2 (N/2) complex multiply and add operations, and which require only N real storage locations in analyzing each N-point record.
134 citations
09 Dec 1968
TL;DR: This article describes another algorithm for computing the Discrete Fourier Transform where the required number of additions and subtractions is the same as in the Cooley-Tukey Algorithm; but therequired number of multiplications is only one half of that in the cooley- Tukey Al algorithm.
Abstract: With the advent of digital computers it became possible to compute the Discrete Fourier Transform for a large number of input points in relatively reasonable times. However, for certain uses a demand developed to compute the Discrete Fourier Transform in a very short time or even in real time. Also, a demand developed for computing the Fourier Transform for a very large number of input points. These demands resulted in a requirement for computing the Fourier Transform in the fastest time possible. A very economical way for computing the Fourier Transform was developed a few years ago and is known as the Cooley-Tukey Algorithm. This article describes another algorithm for computing the Discrete Fourier Transform where the required number of additions and subtractions is the same as in the Cooley-Tukey Algorithm; but the required number of multiplications is only one half of that in the Cooley-Tukey Algorithm.
109 citations
TL;DR: For the complete system of the orthogonal Walsh functions, the implementation of circuits by modem semiconductor techniques appears to be competitive in a number of applications with the implementationof circuits for the system of sine and cosine functions.
Abstract: The system of sine and cosine functions has been distinguished historically in communications. Whenever the term frequency is used, reference is made implicitly to these functions; hence the generally used theory of communication is based on the system of sine and cosine functions. In recent years other complete systems of orthogonal functions have been used for theoretical investigations as well as for equipment design. Analogs to Fourier series, Fourier transform, frequency, power spectra, and amplitude, phase, and frequency modulation exist for many systems of orthogonal functions. This implies that theories of communication can be worked out on the basis of these systems. Most of these theories are of academic interest only. However, for the complete system of the orthogonal Walsh functions, the implementation of circuits by modem semiconductor techniques appears to be competitive in a number of applications with the implementation of circuits for the system of sine and cosine functions.
81 citations
01 Jan 1968
TL;DR: Several properties of the FHT are revealed, including the nature of its presence in the fast Fourier transform, in which it performs the additive operations as shown by further decomposing the product of matrices representing the FFT.
Abstract: : A discrete time transform was studied and applied to the representation and discrimination of digitized signals. The transform consists of an orthogonal (Hadamard) matrix whose elements are all ones and minus ones. To facilitate implementation, a fast Hadamard transform (FHT) has been developed requiring only NlogN rather than N squared algebraic additions. Several properties of the FHT are revealed, including the nature of its presence in the fast Fourier transform, in which it performs the additive operations as shown by further decomposing the product of matrices representing the FFT.
66 citations
TL;DR: In this article, the amplitude spectrum of the Fourier transform is used to estimate the depth to the top of the upper part of a faulted bed and the inclination of the fault-plane.
Abstract: The Fourier transform formula for a two-dimensional fault truncating a horizontal bed at an arbitrary angle of inclination is derived. The amplitude spectrum of the Fourier transform is found to give information about the depth to the top of the upper part of the faulted bed and the inclination of the fault-plane. Under suitable conditions the thickness and the displacement of the bed involved can be obtained. With actual field data, these transforms can be obtained at discrete points by a Fourier analysis of the gravity anomaly. A field example from the Logan fault area near Montreal, Que., Canada, is given.
52 citations
TL;DR: The following procedures are based on the Cooley-Tukey algorithm for computing the finite Fourier transform of a complex data vector; the dimension of the data vector is assumed here to be a power of two.
Abstract: The following procedures are based on the Cooley-Tukey algorithm [1] for computing the finite Fourier transform of a complex data vector; the dimension of the data vector is assumed here to be a power of two. Procedure COMPLEXTRANSFORM computes either the complex Fourier transform or its inverse. Procedure REALTRANSFORM computes either the Fourier coefficients of a sequence of real data points or evaluates a Fourier series with given cosine and sine coefficients. The number of arithmetic operations for either procedure is proportional to n log2n, where n is the number of data points.
32 citations
TL;DR: In this article, the discrete Fourier transform is viewed as a set of discrete linear filters, one filter for each Fourier coefficient, and the characteristics of these filters are discussed.
Abstract: Computational procedures which have been developed in the past few years have taken the familiar frequency-domain techniques from the realm of theory and placed them in the realm of practice. In order to realize fully the potential of th techniques, it is necessary to gain insight into the physical significance of the discrete Fourier transform. Here, the discrete Fourier transform is viewed as a set of discrete linear filters--one filter for each Fourier coefficient. Each filter is seen to have zero poles and (N-1) zeros. (N is the number of data points transformed.) The characteristics of these filters are discussed. Spectrum weighting, for the purpose of sidelobe reduction, is also shown to be equivalent to discrete linear filtering. The filters in this case are similar to those which represent the discrete Fourier transform.
12 citations
01 Sep 1968
TL;DR: Fast Fourier transform (FFT) correlation is compared with direct discrete time (DDT)-based correlation of complex signals in terms of number of operations and memory requirements as mentioned in this paper, and the results establish criteria for evaluating which of the two techniques is the more economical in a real time application involving an indefinitely long input data sequence.
Abstract: Fast Fourier transform (FFT) correlation is compared with direct discrete time (DDT) correlation of complex signals in terms of number of operations and memory requirements. The results establish criteria for evaluating which of the two techniques is the more economical in a real time application involving an indefinitely long input data sequence.
9 citations
TL;DR: In this article, Sine and Cosine Integrals are used to compute the sine and cosine integrals of an Optica Acta, International Journal of Optics: Vol. 15, No. 2, pp. 195-196.
Abstract: (1968). Sine and Cosine Integrals. Optica Acta: International Journal of Optics: Vol. 15, No. 2, pp. 195-196.
01 Sep 1968
TL;DR: The calculation of the optical transfer function from lens design data by digital computer has become common today and describes the imaging of a spatial sine wave by the optical system.
Abstract: The calculation of the optical transfer function from lens design data by digital computer has become common today. The optical transfer function describes the imaging of a spatial sine wave by the optical system. The amplitude of a sine wave is described by the modulation, M, given by the equation© (1968) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: In this paper, it was proved that the sum of the sampled values of the output response, at the sampling interval equal to the period of the input wave, directly yields the sign and cosine transforms respectively.
Abstract: Evaluation of the Fourior transform of the system impulse response is an important aspect of the design of control systems. A method suggested hero requires only one cycle of sine or cosine wave to be applied as an input to the system. It is proved that the sum of the sampled values of the output response, at the sampling interval equal to the period of the input wave, directly yields the sign and cosine transforms respectively. The procedure is generalized to any number of complete cycles of input wave, as well as to n/2 cycles whore n is any odd positive integer.
05 Jun 1968
TL;DR: A computer program is described which determines the Discrete Fourier Transform of a set of data, using a recently developed technique known as the Fast Fourier transform.
Abstract: : This research contribution describes a computer program (CNA Number 76-67) which determines the Discrete Fourier Transform of a set of data, using a recently developed technique known as the Fast Fourier transform. The relation between Discrete Fourier Transforms and Fourier Series when the data is periodic is also known.