Showing papers on "Hartley transform published in 1969"

Hadamard transform image coding

Abstract: The introduction of the fast Fourier transform algorithm has led to the development of the Fourier transform image coding technique whereby the two-dimensional Fourier transform of an image is transmitted over a channel rather than the image itself. This devlopement has further led to a related image coding technique in which an image is transformed by a Hadamard matrix operator. The Hadamard matrix is a square array of plus and minus ones whose rows and columns are orthogonal to one another. A high-speed computational algorithm, similar to the fast Fourier transform algorithm, which performs the Hadamard transformation has been developed. Since only real number additions and subtractions are required with the Hadamard transform, an order of magnitude speed advantage is possible compared to the complex number Fourier transform. Transmitting the Hadamard transform of an image rather than the spatial representation of the image provides a potential toleration to channel errors and the possibility of reduced bandwidth transmission.

An algorithm for computing the mixed radix fast Fourier transform

Abstract: This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey. As in their algorithm, the dimension n of the transform is factored (if possible), and n/p elementary transforms of dimension p are computed for each factor p of n . An improved method of computing a transform step corresponding to an odd factor of n is given; with this method, the number of complex multiplications for an elementary transform of dimension p is reduced from (p-1)^{2} to (p-1)^{2}/4 for odd p . The fast Fourier transform, when computed in place, requires a final permutation step to arrange the results in normal order. This algorithm includes an efficient method for permuting the results in place. The algorithm is described mathematically and illustrated by a FORTRAN subroutine.

Computation of the Fast Walsh-Fourier Transform

Abstract: The discrete, orthogonal Walsh functions can be generated by a multiplicative iteration equation. Using this iteration equation, an efficient Walsh transform computation algorithm is derived which is analogous to the Cooley-Tukey algorithm for the complex-exponential Fourier transform.

The finite Fourier transform

Abstract: The finite Fourier transform of a finite sequence is defined and its elementary properties are developed. The convolution and term-by-term product operations are defined and their equivalent operations in transform space are given. A discussion of the transforms of stretched and sampled functions leads to a sampling theorem for finite sequences. Finally, these results are used to give a simple derivation of the fast Fourier transform algorithm.

A Linear Filtering Approach to the Computation of the Discrete Fourier Transform

Abstract: This chapter contains sections titled: Introduction, An Algorithm Suggested by Chirp Filtering

Theory and Implementation of the Discrete Hilbert Transform

Abstract: The Hilbert transform has traditionally played an important part in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform plays a similar role in digital signal processing. In this paper, the Hilbert transform relations, as they apply to sequences and their z-transforms, and also as they apply to sequences and their Discrete Fourier Transforms, will be discussed. These relations are identical only in the limit as the number of data samples taken in the Discrete Fourier Transforms becomes infinite. The implementation of the Hilbert transform operation as applied to sequences usually takes the form of digital linear networks with constant coefficients, either recursive or non-recursive, which approximate an all-pass network with 90° phase shift, or two-output digital networks which have a 90° phase difference over a wide range of frequencies. Means of implementing such phase shifting and phase splitting networks are presented.

Application of Walsh Functions to Transform Spectroscopy

Abstract: THIS article suggests that Walsh functions1–3 might be used in transform Spectroscopy4–6 to replace the sinusoidal functions appearing in the Fourier transform. We think this might be the case because Walsh functions are a complete orthonormal set, and therefore give rise to an integral transform of Fourier type; and they take only the values + 1 and − 1 and are therefore likely to be well suited to the binary digital computer.

Some applications for Kramer's generalized sampling theorem

Abstract: Kramer's generalization of Shannon's sampling theorem takes us from a signal represented by a finite Fourier transform to a signal represented by another and more general finite integral transform. In this paper we will attempt to show that the already obtained results for Kramer's theorem are of use in the field of finite integral transforms. Also by introducing such transforms one can treat some communications problems. An example is the case of representing a signal which is the output of time variant filter.

Application of a Modified Fast Fourier Transform to Calculate Human Operator Describing Functions

Abstract: A modified fast Fourier transform (FFT) is used in a hybrid computer program to permit processing of tracking data during a run to yield the human operator's describing function almost immediately after the data-taking period. The computer processing time is substantially reduced at no cost in accuracy.

Comments on "A High-Speed Algorithm for the Computer Generation of Fourier Transforms"

Abstract: The efficiency (in terms of both execution time and storage requirements) of a recently presented algorithm for computing the fast Fourier transform is compared to that of alternative algorithms.

Analytic Implementation of Laplace Transform Algorithms Using the Digital Computer

Abstract: The Laplace transform of a function can be recognized as the result of a finite number of operations perforned on the transform of a simple function. The rules governing the permissible sequences of operation have been established previously. The resultant algorithms have been used to teach Laplace transform techniques and have been applied in the derivation of transforms. From these algorithms a computer program has been developed to transform analytically from the t to the s domain. A brief survey of the algorithms is followed by a description of the program logic and grammar. Examples are given to illustrate the capabilities and limitations of the program. Possible applications and extensions of the program are discussed.