Showing papers on "Hartley transform published in 1974"

Fourier Transform Computers Using CORDIC Iterations

Abstract: The CORDIC iteration is applied to several Fourier transform algorithms. The number of operations is found as a function of transform method and radix representation. Using these representations, several hardware configurations are examined for cost, speed, and complexity tradeoffs. A new, especially attractive FFT computer architecture is presented as an example of the utility of this technique. Compensated and modified CORDIC algorithms are also developed.

Fast Convolution using fermat number transforms with applications to digital filtering

Abstract: The structure of transforms having the convolution property is developed. A particular transform is proposed that is defined on a finite ring of integers with arithmetic carried out modulo Fermat numbers. This Fermat number transform (FNT) is ideally suited to digital computation, requiring on the order of N \log N additions, subtractions and bit shifts, but no multiplications. In addition to being efficient, the Fermat number transform implementation of convolution is exact, i.e., there is no roundoff error. There is a restriction on sequence length imposed by word length but multi-dimensional techniques are discussed which overcome this limitation. Results of an implementation on the IBM 370/155 are presented and compared with the fast Fourier transform (FFT) showing a substantial improvement in efficiency and accuracy.

Canonical transforms. I. Complex linear transforms

Abstract: Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x and p to a pair η and ζ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η, ζ can be realized in the Schrodinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square‐integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well‐known Bargmann transform is the special case when η and iζ are the harmonic oscillator raising and lowering operators. The Moshinsky‐Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.

Computer-generated Fourier transforms of helical particles

Abstract: An alternative method has been found to display the information contained in the Fourier transform of a helical particle. This allows a strong selection rule to be defined for the transform on the layer lines and consequently the discrimination between signal and noise contributions to the data can be improved.

The power spectrum of the Mellin transformation with applications to scaling of physical quantities

Abstract: The Mellin transform is used to diagonalize the dilation operator in a manner analogous to the use of the Fourier transform to diagonalize the translation operator. A power spectrum is also introduced for the Mellin transform which is analogous to that used for the Fourier transform. Unlike the case for the power spectrum of the Fourier transform where sharp peaks correspond to periodicities in translation, the peaks in the power spectrum of the Mellin transform correspond to periodicities in magnification. A theorem of Wiener‐Khinchine type is introduced for the Mellin transform power spectrum. It is expected that the new power spectrum will play an important role extracting meaningful information from noisy data and will thus be a useful complement to the use of the ordinary Fourier power spectrum.

Two-dimensional digital processing of one-dimensional signal

Abstract: It is of importance to find the necessary and sufficient conditions under which the one-dimensional and two-dimensional processing of any general transform should be equivalent. These conditions are found. It is known that the Fourier transform does not possess the described property. It is shown in this paper that a one-dimensional Fourier transform cannot be equivalent to any two-dimensional transform. On the other hand it is shown that the two-dimensional Fourier transform is equivalent to a one-dimensional transform of another kind and that the processing can be performed by a fast algorithm.

A class of generalized continuous orthogonal transforms

Abstract: This paper presents a class of generalized continuous transforms for the orthogonal decomposition of signals. Base functions for the continuous transform range from Walsh functions of order two to stair-like functions which resemble approximations to sinusoids and which are distinct from the generalized Walsh functions. Standard desirable properties which are shown to hold for the generalized continuous transform operator include orthogonality of the base functions, linearity of the transform operator, inverse transformability, and admissibility to fast transform representation. The transform class is governed by a definition of time translation in terms of signed-bit dyadic time shift. Mathematical properties leading to this definition are discussed and the impact of the definition is assessed. Properties of the continuous class of generalized transforms make feasible analysis which could be extremely tedious using matrix representations of the operations actually mechanized in a sampled-data system. Analysis techniques are illustrated with a target detection system which is conceptually designed using the generalized continuous transform and implemented using fast transform algorithms to perform correlation operations. Since the correlation operations are valid for inputs which include signals represented in terms of Walsh functions, the example illustrates one instance in which the binary Fourier representation (BIFORE) transform can be used for practical pattern recognition.

An improved fast Fourier transform

Abstract: This correspondence presents an improved implementation of the fast Fourier transform without sorting.

Integral transform functions and separable potentials

Abstract: General forms for asymptotic wave functions are derived from properties of the relevant Green's function. The use of separable potentials constructed from the asymptotic functions is described and the relation with integral transform functions is discussed.

Fourier Transform and Differential Equations

Abstract: The Fourier transform is one of the most powerful tools in classical and modern analysis. Its scope has recently been strikingly extended thanks to the introduction of the notion of generalized functions of S. L. Sobolev [1] and L. Schwartz [1]. The extension has been applied successfully to the theory of linear partial differential equations by L. Ehrenpreis, B. Malgrange and especially by L. Hormander [6].

Fourier Transform Methods for Solution of Differential Equations

Abstract: The use of Fourier transform and of the equivalent Laplace transform methods for the solution of differential equations with constant coefficients is often preferred to symbolic methods because of a belief that they are more rigorous. While the transform methods are easier to apply correctly, because the rules for correct application can be stated in terms of classical analysis, the degree to which they are rigorous tends to be overestimated. Logically, the transform methods proceed as follows. To solve a problem P concerning an unknown function y, one forms the transform Ty of the function y by means of a linear transformation T acting on a space of functions E; and one finds a transformed problem say TP obeyed by Ty if y obeys P. A solution Y of the problem TP is found, and then by inversion procedures the function y in the original space E corresponding to Y is calculated.