Showing papers on "Non-uniform discrete Fourier transform published in 1977"

A Fast Computational Algorithm for the Discrete Cosine Transform

Abstract: A Fast Discrete Cosine Transform algorithm has been developed which provides a factor of six improvement in computational complexity when compared to conventional Discrete Cosine Transform algorithms using the Fast Fourier Transform. The algorithm is derived in the form of matrices and illustrated by a signal-flow graph, which may be readily translated to hardware or software implementations.

Short term spectral analysis, synthesis, and modification by discrete Fourier transform

Abstract: A theory of short term spectral analysis, synthesis, and modification is presented with an attempt at pointing out certain practical and theoretical questions. The methods discussed here are useful in designing filter banks when the filter bank outputs are to be used for synthesis after multiplicative modifications are made to the spectrum.

Quasi fast Hankel transform.

Abstract: We outline here a new algorithm for evaluating Hankel (Fourier–Bessel) transforms numerically with enhanced speed, accuracy, and efficiency. A nonlinear change of variables is used to convert the one-sided Hankel transform integral into a two-sided cross-correlation integral. This correlation integral is then evaluated on a discrete sampled basis using fast Fourier transforms. The new algorithm offers advantages in speed and substantial advantages in storage requirements over conventional methods for evaluating Hankel transforms with large numbers of points.

An introduction to programming the Winograd Fourier transform algorithm (WFTA)

Abstract: Recently, Dr. Shmuel Winograd discovered a new approach to the computation of the discrete Fourier transform (DFT). Relative to fast Fourier transform (FFT), the Winograd Fourier transform algorithm (WFTA) significantly reduces the number of multiplication operations; it does not increase the number of addition operations in many cases. This paper introduces the new algorithm and discusses the operations comparison problem. A guide for programming is included, as are some preliminary running times.

Video-Rate Image Correlation Processor

Abstract: A digital processor has been designed and built to implement Lockheed's Phase Correlation technique at a rate of 30 correlations per second on 128 x 128 element images digitized to eight bits. Phase Correlation involves taking the inverse Fourier transform of the appropriately filtered phase of the Fourier cross-power spectrum of a pair of images to extract their relative displacement vector. It achieves sub-pixel accuracy with relative insensitivity to scene content, illumination differences and narrow-band noise. The processor, which is designed to accept inputs from a variety of sensors, is built with conventional TTL and MOS components and employs only a moderate amount of parallelism. It uses floating point arithmetic with equal exponents for real and imaginary parts. Multiplications are performed by table lookup. Application areas for the correlator include image velocity sensing, correlation guidance and scene tracking.© (1977) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Efficient structure-factor calculation for large molecules by the fast Fourier transform

Abstract: A method is presented for calculating structure factors by Fourier inversion of a model electron density map. The cost of this method and of the standard methods are analyzed as a function of number of atoms, resolution, and complexity of space group. The cost functions were scaled together by timing both methods on the same problem, with the same computer. The FFT method is 3½ to 7 times less expensive than conventional methods for non-centrosymmetric space groups.

Direct fast Fourier transform of bivariate functions

Abstract: A mathematical development is presented for a direct computation of a two-dimensional fast Fourier transform (FFT). A generalized mathematical theory of holor algebra is used to manipulate coefficient arrays needed to generate computational equations. The result is a set of equations which involve elements from throughout the two-dimensional array rather than operating on individual rows and columns. Preliminary digital computer calculations verify the accuracy of the technique and demonstrate a modest saving of computation time as well.

Channel equalization apparatus and method using the Fourier transform technique

Abstract: A method of and apparatus for determining during an initial training period the initial values of the coefficients of a transversal equalizer in a data transmission system in which the transmission channel creates frequency shift. The received periodic training sequence is modulated by a time-domain window signal whose Fourier transform exhibits a relatively flat central peak and has comparatively low values in the vicinity of those frequencies which are a multiple of the inverse of the period of the transmitted sequence, and the discrete Fourier transform Wk of the modulated signal is computed. The values of the coefficients of the equalizer are obtained by computing the inverse discrete Fourier transform of the ratio Fk =Zk /Wk, where Zk is the discrete Fourier transform of the transmitted sequence.

The zoom FFT using complex modulation

Abstract: A recent paper by Yip discussed the zoom transform as derived from the defining equation of the FFT. This paper simplifies the concepts and removes some of the restrictions assumed by Yip; ie., the total number of points need not be a power of 2. The technique is based on first specifying the desired center frequency, bandwidth, and frequency resolution. The signal is then sampled, modulated, and lowpass filtered. This result is purposely aliased, then transformed using an FFT algorithm. The result is an M-point frequency spectra of the desired bandwidth centered about the center frequency with a higher degree of resolution than could be directly obtained using an M-point transform.

Image Processing by Transforms Over a Finite Field

Abstract: A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.

Digital filtering using pseudo fermat number transforms

Abstract: In this paper pseudo Fermat number transforms (FNT's) are discussed. These transforms are defined in a ring of integers modulo an integer submultiple of a pseudo Fermat number, and can be computed without multiplications while allowing a great flexibility in word length selection. Complex pseudo FNT's are then introduced and are shown to relieve some of the length limitations of conventional Fermat number transforms (FNT's). These transforms, which under certain conditions can be computed via fast transform algorithms allow the implementation of digital filters with better efficiency and accuracy than the fast Fourier transform (FFT).

Fourier transformation of rotationally invariant two-variable functions: Computer implementation of Hankel transform

Abstract: Computing the Fourier transform of a circularly symmetric function is often necessary in optics. Use of the 2-D FFT algorithm leads to loss of the symmetry because of the sampling and to a waste in storage requirements; to avoid these inconveniences, a 1-D algorithm is described using the Hankel transform of the section of the function.

A Differential Technique for the Fourier Transform Processing of Multicomponent Exponential Functions

Abstract: Conventional Fourier transform processing of multicomponent exponential functions of the form f(t) = ?iAie-?it commences by forming the product exf(ex). This note is concerned with an alternative starting point?the formation of the x-derivative, f'(ex). It is indicated that this approach: 1) yields processed outputs whose peaks are proportional to Ai directly; 2) offers an advantageously different noise performance; 3) can deal with functions containing a constant (D. C.) bias level; and 4) requires in practice only the formation of first differences.

'Instant' Fourier transform

Abstract: The fast Fourier transform andd the fast Walsh transform are too slow for some real-time applications. For binary data, an 'instant' Fourier transform is based on harmonic analysis in a space of 2 n -tuples of 0s and 1s. Simple, modular logic finishes transforming 2n real-time serial binary data one clock pulse after the last datum arrives.

Analysis of the gravity effect of two‐dimensional trapezoidal prisms using fourier transform*

Abstract: The gravity effect of an infinite horizontal trapezoidal prism is derived and its Fourier spectrum is analyzed so as to yield information about four parameters of the causative structure, namely the depths to the upper and lower surfaces, width of the upper surface, and the inclination of the sides. In order to test the applicability of the method, synthetic data are constructed by digitizing the theoretical gravity effect. Subsequently, the corresponding Discrete Fourier Transform (DFT) is obtained. The parameters evaluated from the DFT are observed to be sufficiently close to the chosen values.

The Beatquency Domain: An Unusual Application of the Fast Fourier Transform

Abstract: Fourier analysis has proven to be a vital mathematical tool in many areas of research, but rapid methods for calculating frequency content of sampled data using discrete Fourier transform (FFT) require periodic sampling. Unfortunately, beat-by-beat heart rate is an aperiodic series of events in the time domain. This report describes a new and unusual application of the FFT on heart rate data. The beat-by-beat intervals are represented as the magnitude of a periodically sampled function. When the Fourier transform is applied to these data, we obtain pseudofrequency information from what we call the Beat-quency Domain. Observed patterns from the Beatquency Domain suggest usefulness of this method in sleep cycle detection.

An algorithm for the two dimensional FFT

Abstract: Conventional two dimensional fast Fourier transforms become very slow if the size of the matrix becomes too large to be contained in memory. This is due to the transposition of the matrix that is required. This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process. This algorithm is extremely valuable on small disc based computers.

A method for programming the complex general-N Winograd Fourier transform algorithm

Abstract: The Winograd Fourier Transform Algorithm (WFTA) requires about 20% of the multiplications used in an optimized FFT, while the number of additions remains unchanged. This paper describes one "General-N" (i.e. many allowable DFT sizes (N) but certainly not any vector size) complex WFTA programming technique.

Fast two dimensional fourier transform device

Abstract: Two dimensional optical or electrical images are processed through a storage tube designed to yield the correlation function between the input images and stored images. By generating a series of stored images representing sine and cosine components of the Fourier transform, a fast, two-dimensional transform of the image is obtained.

A Table of Discrete Fourier Transform Pairs

Abstract: A classification of methods for generating discrete Fourier transform pairs is given, followed by a table of 29 pairs. Many of these are new, whereas some have been collected from various literature sources. We have tried to make the table interesting rather than comprehensive. The generalization of the Gaussian sums is a good example. Hundreds of additional nonobvious finite identities can be deduced by using the Rayleigh-Parseval formula and convolutions.

The Sinc and Cosinc Transform

Abstract: A new transform, the sinc and cosinc transform, uses Walsh functions to obtain the Fourier transform. This technique converts a staircase approximation of a function to a set of sinc and cosinc terms in the frequency domain that is equivalent to the Fourier transform. The calculation is slower than the fast Fourier transform (FFT) but is devoid of aliasing. The interpolation and scaling in the frequency domain are built in, and any frequency point may be chosen without changing the number or spacing of the samples in the time domain. The intervening set of coefficients is computed more rapidly than those obtained using the fast Hadamard transform.

Winograd's algorithm applied to number-theoretic transforms

Abstract: We show how to perform a number-theoretic transform (n.t.t.) using an algorithm analogous to that of S. Winograd for computing the discrete Fourier transform (d.f.t.). Using this algorithm, the range of data lengths and word lengths is much larger than that available with conventional fast n.t.t.s.

A two-pass fixed point fast Fourier transform error analysis

Abstract: A statistical model is used to predict the output signal-to-noise ratio (SNR) when a two-pass fast Fourier transform (FFT) is computed using fixed-point arithmetic. The results show that the ratio varies essentially as the square root of the number of points in the transform. Also included are the results of the simulation of a fixed-point machine and the variation of the error as a function of the length of the coefficients.

Realization of a discrete Fourier transform (DFT) module for incorporation in FFT processors

Abstract: For applications requiring high-speed and in-place treatment, it is often advantageous to realize special-purpose computers. This paper describes a discrete Fourier transform (DFT) module for incorpration in fast Fourier transform (FFT) processors. The module is highly suitable for real input applications requiring high-speed transformations. It attributes one point to all frequency channels in one clock cycle. This treatment is not only well suited for the present technology, but appears to be more attractive in view of recent trends in digital circuitry.

Rapid computation of Fourier texture descriptors

Abstract: Computer vision systems based on general purpose computers often need efficient texture description algorithms. One common method is to calculate two-dimensional discrete Fourier transforms over windowed regions of the light intensity matrix. Although these methods described in the literature are based on the fast Fourier transform algorithm, the computation time is still too high to permit the description of texture for as many windows as are needed for good segmentation. When a set of transforms over a window at every position of the matrix is needed, an efficient method can be used. It saves information computed for previous windows and uses it to reduce the effort expended on the current window. For a window N × N and an image matrix M × M, the time complexity is reduced from O(N2M2logN) to O(N2M2). This complexity cannot be beaten by any nonparallel algorithm.

Examples Of Maximum Likelihood Spatial Filtering

Abstract: SUMMARY We describe a maximum likelihood method for estimating spatial frequency components of truncated sections of data. The method is used to estimate the onedimensional Fourier transform of short scans of images of an edge and of a slit. With the aid of a constrained least squares noise control, t he frequency response of the imaging system is computed from the estimated Fourier transform. The application of Fourier transform techniques to image processing usually requires large amounts of data. Much of spatial filtering theory is based on the Fourier integral with its infinite limits, which we must approximate in practice with a finite integration region. Images must be truncated, and the truncation leads to artifacts in the processed image. These tend to be most prominent near the edges, and, in practice, we often discard all but t he c enter r egion of the image, where errors are smaller.