Showing papers on "Fractional Fourier 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.

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.

On fourier’s analysis of linear inequality systems

Abstract: Fourier treated a system of linear inequalities by a method of elimination of variables. This method can be used to derive the duality theory of linear programming. Perhaps this furnishes the quickest proof both for finite and infinite linear programs. For numerical evaluation of a linear program, Fourier’s procedure is very cumbersome because a variable is eliminated by adding each pair of inequalities having coefficients of opposite sign. This introduces many redundant inequalities. However, modifications are possible which reduce the number of redundant inequalities generated. With these modifications the method of Fourier becomes a practical computational algorithm for a class of parametric linear programs.

Coarse frequency estimation using the discrete Fourier transform (Corresp.)

Abstract: The discrete Fourier transform (DFT) is applied as a coarse estimator of the frequency of a sine wave in Gaussian noise. Probability of anomaly and the variance of the estimation error are determined by computer simulation for several DFT block sizes as a function of signal energy-to-noise density ratio \mathcal{E}/N_0 . Several data windows are considered, but uniform weighting gives the best performance.

Diffraction calculations using fast Fourier transform methods

Abstract: By using a coordinate transformation based on Gaussian-beam theory, we can apply efficient fast Fourier transform (FFT) computational methods to diffraction problems involving spherically diverging or converging waves as well as to quasi-collimated beams of radiation.

Method and apparatus for computing the discrete Fourier transform recursively

Abstract: A wholly digital system for computing the discrete Fourier transform of sequentially received data in a recursive fashion. Two parallel shift registers store and shift the real and imaginary components of the complex number X k + iY k . The data in the parallel registers are successively shifted one bit per strobe in response to receipt of new data. Additional logic operates recursively on successive data inputs to compute the discrete Fourier transform.

Implementation of a fast Fourier transform (FFT) for image processing applications

Abstract: Different fast Fourier transform (FFT) algorithms for hardware implementation have been considered. We propose an implementation whereby two radix-N1/2passes are carried out in parallel and in which each N1/2-point transform is carried out via a serial input parallel output transform circuit. The processing rate is one clock cycle per input point for the N-point transform regardless of the value of N chosen. The circuit is being implemented with TTL logic and will be used to perform spatial frequency domain filtering on two dimensional infrared camera images in real time; real time meaning processing between frame display.

Frame-to-frame coding of television pictures using two-dimensional Fourier transforms (Corresp.)

Abstract: Two systems are described for reducing the data rate required to transmit TV signals. Basic to the operation of both systems is the fact that if there is an object in the scene moving more or less in linear translation, then the two-dimensional frame-difference signal computed during the present frame will be very similar to the frame-difference signal which was computed during the previous frame except for a linear translation. Thus the Fourier transform of the frame-difference signal will also be similar during successive frames except for a linear phase shift.

Comparative Study of a Discrete Linear Basis for Image Data Compression

Abstract: Transform image data compression consists of dividing the image into a number of nonoverlapping subimage regions and quantizing and coding the transform of the data from each subimage. Karhunen-Loeve, Hadamard, and Fourier transforms are most commonly used in transform image compression. This paper presents a new discrete linear transform for image compression which we use in conjunction with differential pulse-code modulation on spatially adjacent transformed subimage samples. For a set of thirty-three 64 × 64 images of eleven different categories, we compare the performancea of the discrete linear transform compression technique with the Karhunen-Loeve and Hadamard transform techniques. Our measure of performance is the mean-squared error between the original image and the reconstructed image. We multiply the mean-squared error with a factor indicating the degree to which the error is spatially correlated. We find that for low compression rates, the Karhunen-Loeve outperforms both the Hadamard and the discrete linear basis method. However, for high compression rates, the performance of the discrete transform method is very close to that of the Karhunen-Loeve transform. The discrete linear transform method performs much better than the Hadamard transform method for all compression rates.

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.

The shadow transform: An approach to cross sectional imaging

Abstract: : The goal of this work is to define and investigate the properties of the linear operator that relates the density to its set of projections--the shadow. Several approaches to the definition of the shadow transform are presented in Chapter I. Chapter II provides the basis for the spectral analysis by deriving the eigenfunctions of the shadow transform. Chapter III deals with the derivation of reconstruction schemes (realizations of the inverse shadow transform). In Chapter IV the author develops the theory for the case where the measurement of the projections is limited by the geometry.

Motion detection employing direct Fourier transforms of images

Abstract: Method and apparatus for directly converting between an image and the spatial or temporal Fourier transform thereof. To convert an image into its Fourier transform representation, the image interacts with strain waves in media that have electrical properties varying as a function of both the intensity pattern of the image and strain waves in the media. The electrical properties are measured to derive signals representing Fourier series terms defining the image. The derived signals are used to detect motion (including motion in the plane of the image), for image stabilization and scaling, and for pattern recognition. A new DEFT device (Direct Electronic Fourier Transform) obtains a Fourier transform representation of an image by utilizing photon assisted tunnelling current through an isolator film junction between two thin conductor films. Another new DEFT device provides spatial scanning similar to television raster scanning but utilizing completely different principles. Still another new DEFT device generates a two-dimensional spatial Fourier transform representation of an image without the need for two-dimensional scanning of the strain wave. An image is reconstructed from electrical signals obtained as described above by interacting uniform (but not necessarily coherent) light with strain waves that are a function of these electrical signals.

Direct application of the fast fourier transform to open resonator calculations.

Abstract: It is shown how fast Fourier transform techniques can be used to efficiently, numerically evaluate (convolution) integrals of the form often encountered in open resonator eigenmode calculations.

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.

Generation of Dolph-Chebyshev weights via a fast Fourier transform

Abstract: Dolph-Chebyshev weights, which realize a minimum side-lobe level for a specified main-lobe width, can be generated by a single fast Fourier transform (FFT). For an even number of elements 2H, the size of the FFT is H. This result has utility for spectral analysis as well as for array processing.

A simple formalism for the study of transformed aeromagnetic profiles and source location problems

Abstract: Up to now, magnetic interpretation through processing of aeromagnetic maps has been based on two classes of methods, Fourier transform and direct convolution techniques; the first class requires transformation of entire maps and does not easily allow local studies, whereas the other gives only numerical values of filter coefficients. In the present paper a suitable continuation function is determined and used to obtain simple analytical expressions for all classical filters (upward and downward continuation, vertical gradient and derivatives up to any order, and reduction to the pole). This formalism allows local studies to be performed and gives both physical and mathematical insight into transformed aeromagnetic profiles; at the same time it leads to simple formulas and cheap digital computations and should be handled easily by nonspecialists. Tests on analytical and numerical models show the quality of the continuation function and of the various transformed profiles. It is also shown that study of the behavior of the continuation function can yield valuable information to locate sources of magnetic anomalies with good precision.

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.

Sequency Domain Design of Frequency Filters

Abstract: Each discrete transform has its advantages over other transforms. Fourier domain filters are easier to design than Walsh domain filters. But Walsh transforms can be computed much faster than Fourier transforms. This paper considers the problem of designing a Walsh domain filter, given a Fourier domain filter. Same procedure can be applied to other transforms. Given a Slant filter its equivalent Fourier filter or Walsh ifiter can be found.

Fourier Transforms of Walsh Functio ns

Abstract: The Z-transform of a discrete square-wave function simplifies the derivation of an expression for the Fourier transform of a continuous square-wave function defined over a finite interval. In particular, since the Z-transform of any Walsh function can be written in closed form by inspection of the G ray code representation of its index, the Fourier transform can be readily written down as a trigonometric product function.

An algorithm for performing an inverse chirp z-transform

Abstract: An efficient algorithm for the determination of the coefficients of a polynomial from evenly spaced sample values of that polynomial on a spiral contour in the complex plane is presented. It is useful for the determination of a sequence from evenly spaced values of its Fourier transform.

Parallel data streams and serial arithmetic for fast Fourier transform processors

Abstract: An algorithm is presented that introduces two degrees of parallelism into the implementation of fast Fourier transform (FFT) processors. That is, both the radix of factorization and the number of arithmetic units may be selected to achieve the required processing speed. A serial vector multiplier that is ideally suited to the implementation of a general radix arithmetic unit is described. It is subsequently shown that a fast Fourier processor having an attractive cost performance ratio can be built by employing serial arithmetic in the implementation of the algorithm developed.