Showing papers on "Fast Fourier transform published in 1983"

Discrete Hartley transform

Abstract: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ HartleyR. V. L., Proc. IRE30, 144 ( 1942)] as the DFT bears to the Fourier transform.

A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation

Abstract: From the standpoint of reporting a new contribution, this paper shows that by using bilinear interpolation followed by direct two-dimensional Fourier inversion, one can obtain reconstructions of quality which is comparable to that produced by the filtered-backpropagation algorithm proposed recently by Devaney. For an N × N image reconstructed from N diffracted projections, the former approach requires approximately 4N FFT's, whereas the backpropagation technique requires approximately N2FFT's. We have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography, an area that is becoming increasingly important not only in medical imaging, but also in underwater and seismic mapping with microwaves and sound. The main feature of the tutorial part is the statement of the Fourier diffraction projection theorem, which is an extension of the traditional Fourier slice theorem to the case of image formation with diffracting illumination.

Fast Transforms Algorithms, Analyses, Applications

Abstract: Preface. Acknowledgments. List of Acronyms. Notation. Introduction. Fourier Series and Fourier Transform. Discrete Fourier Transforms. Fast Fourier Transform Algorithms. FFT Algorithms That Reduce Multiplications. DFT Filter Shapes and Shaping. Spectral Analysis Using the FFT. Walsh-Hadamard Transforms. The Generalized Transform. Discrete Orthogonal Transforms. Number Theoretic Transforms. Appendix. References. Index.

Fourier Transforms in VLSI

Abstract: This paper surveys nine designs for VLSI circuits that compute N-element Fourier transforms. The largest of the designs requires O(N2 log N) units of silicon area; it can start a new Fourier transform every O(log N) time units. The smallest designs have about 1/Nth of this throughput, but they require only 1/Nth as much area.

A fast Kalman filter for images degraded by both blur and noise

Abstract: In this paper a fast Kalman filter is derived for the nearly optimal recursive restoration of images degraded in a deterministic way by blur and in a stochastic way by additive white noise. Straightforwardly implemented optimal restoration schemes for two-dimensional images degraded by both blur and noise create dimensionality problems which, in turn, lead to large storage and computational requirements. When the band-Toeplitz structure of the model matrices and of the distortion matrices in the matrix-vector formulations of the original image and of the noisy blurred observation are approximated by circulant matrices, these matrices can be diagonalized by means of the FFT. Consequently, a parallel set of N dynamical models suitable for the derivation of N low-order vector Kalman filters in the transform domain is obtained. In this way, the number of computations is reduced from the order of O(N4) to that of O(N^{2} \log_{2} N) for N × N images.

A New Implementation of the Mellin Transform and its Application to Radar Classification of Ships

Abstract: A modified Mellin transform for digital implementation is developed and applied to range radar profiles of naval vessels. The scale invariance property of the Mellin transform provides a means for extracting features from the profiles which are insensitive to the aspect angle of the radar. Past implementations of the Mellin transform based on the FFT have required exponential sampling, interpolation, and the computation of a correction term, all of which introduce errors into the transform. In addition, exponential sampling results in a factor of ln N increase in the number of data points. An alternate implementation, developed in the paper, utilizes a direct expansion of the Mellin integral definition. This direct Mellin transform (DMT) eliminates the implementation problems associated with the FFT approach, and does not increase the number of samples. A scale and translation invariant transform is developed from a modification of the DMT. The MDMT applied to the FFT of the radar profiles results in the desired insensitivity without having the low-pass filtering characteristic that exists in other Fourier-Mellin implementations.

Sign/Logarithm Arithmetic for FFT Implementation

Abstract: Sign/logarithm arithmetic is applicable to a variety of numerical applications where wide dynamic range and small wordsize are required. In this paper the basic sign/logarithm arithmetic operations required for signal processing (i.e., addition, subtraction, and multiplication) are reviewed, the computational errors are analyzed for FFT realization, and simulation results are presented which serve to verify the analysis. It is shown that the sign/logarithm approach provides improved arithmetic quantization error performance for a given word size over FFT's implemented with conventional fixed or floating point arithmetic, and that the sign/logarithm implementation is faster and less complex than conventional approaches.

Computation of Faber series with application to numerical polynomial approximation in the complex plane

Abstract: Kovari and Pommerenke [19], and Elliott [8], have shown that the truncated Faber series gives a polynomial approximation which (for practical values of the degree of the polynomial) is very close to the best approximation. In this paper we discuss efficient Fast Fourier Transform (FFT) and recursive methods for the computation of Faber polynomials, and point out that the FFT method described by Geddes [13], for computing Chebyshev coefficients can be generalized to compute Faber coefficients. We also give a corrected bound for the norm of the Faber projection (that given in Elliott [8], being unfortunately slightly in error) and very briefly discuss a possible extension of the method to the case when the mapping function, which is required to compute the Faber series, is not known explicitly.

Optimal smoothing of 'noisy' data by fast Fourier transform

Abstract: The theory of optimal filtering and smoothing of noisy data is presented. Implementation of this theory is made on the AS/7000 computer at Daresbury Laboratory. The Fortran code and examples of application on 'typical' spectroscopic data are given. The routine size is 35 kbytes and the CPU time 0.11 s for 1024 points.

Practical Algorithms for Mean Velocity Estimation in Pulse Doppler Weather Radars Using a Small Number of Samples

Abstract: Doppler weather radars with fast scanning rates must estimate spectral moments based on a small number of echo samples. This paper concerns the estimation of mean Doppler velocity in a coherent radar using a short complex time series. Specific results are presented based on 16 samples. A wide range of signal-to-noise ratios are considered, and attention is given to ease of implementation. It is shown that FFT estimators fare poorly in low SNR and/or high spectrum-width situations. Several variants of a vector pulse-pair processor are postulated and an algorithm is developed for the resolution of phase angle ambiguity. This processor is found to be better than conventional processors at very low SNR values. A feasible approximation to the maximum entropy estimator is derived as well as a technique utilizing the maximization of the periodogram. It is found that a vector pulse-pair processor operating with four lags for clear air observation and a single lag (pulse-pair mode) for storm observation may be a good way to estimate Doppler velocities over the entire gamut of weather phenomena.

A spectral-iteration technique for analyzing scattering from arbitrary bodies, Part I: Cylindrical scatterers with E-wave incidence

Abstract: In the past, methods for solving electromagnetic scattering problems in the frequency domain have been developed largely for the low-frequency (moment method) and high-frequency (asymptotic techniques) regimes. The intermediate frequency range has been analyzed by combinations of these two approaches or by separation of variables, when possible. This paper is devoted to the development of an independent approach, viz., the "stacked spectral-iteration technique," which is capable of handling arbitrary scatterers with dimensions ranging from small to moderately large. The method takes advantage of the simplicity with which the planar-source planar-field relationships are expressed in the spectral domain. The boundary conditions or constitutive relationships, on the other hand, are expressed most simply in the spatial domain. Alternating between the two domains is carried out with the aid of the fast Fourier transform (FFT) algorithm. The spectral-iteration technique (SIT) was applied in the past to thin planar structures which allow the analysis to be carried out on a plane. The generalization of the two-dimensional formulation to arbitrary three-dimensional bodies can be accomplished by sampling the current distribution on the scatterer over a number of parallel planes, and using the simple spectral-domain interaction relationships between the planes. This new approach involves no matrix inversion and is capable of analyzing scatterers whose sizes far exceed those treatable by the moment method. In addition to being arbitrarily shaped, the scatterer may be conducting, dielectric, or lossy dielectric. Thus, the SIT provides an efficient approach to filling the much-needed gap between low- and high-frequency conventional techniques, e.g., the moment method (MoM) and the geometrical theory of diffraction (GTD), and to extending the range of applicability to dielectric scatterers, with or without loss. Though the concepts presented herein are applicable to arbitrary three-dimensional scatterers, the problem of arbitrary cylinders with E -polarized excitation is addressed in this paper, while the H -case is treated in an accompanying work. The three-dimensional case is to be reported in a future communication which treats the problem of scattering by a lossy inhomogeneons dielectric cylinder of finite length.

Spectrum display device for audio signals

Abstract: An input audio signal is AD converted into digital signals which are processed by a central processing unit (CPU) in which Fast Fourier Transform (FFT) operation and power spectrum calculation are effected. As a result power spectrum data is obtained, and then position determination is effected to determine the position of each bar of a bar graph to be displayed on a display unit screen, according to the power level at different frequencies within the spectrum of the audio signal. Pattern data is then produced in correspondence with the determined position, and output data from the CPU is fed via a video display processor to a video RAM, thereby displaying the spectrum by way of a predetermined pattern of a bar graph on the screen. In order to reduce the number of digital data used in FFT operation the input audio signal may be divided into a plurality of different frequency bands so that different sampling frequencies are used for AD conversion of signals of respective bands. As a result, operating time is reduced. To further reduce the operating time the CPU may be arranged to execute parallel operations in various manners. Furthermore, two or more CPUs may be employed to further increase the operating speed.

Universal electric motor speed sensing by using Fourier transform method

Abstract: An indirect speed sensor for an electric motor, such as an electric motor of the universal type, provides for a resistive element in series with the motor windings so that current pertubations that occur as a consequence of armature rotation are manifested as speed-dependent voltage signals across the resistive element. The voltage signals are amplified and filtered to remove frequency components above and below the expected signal spectrum and then provided to an analog-to-digital converter that converts the filtered analog signal to digital values that are stored in a memory. A stored-program controlled microprocessor analyzes the stored digital information by performing a fast Fourier transform to provide an equivalent amplitude spectrum. The maximum amplitude spectral component is then identified to determine the speed of the motor.

The design of optimal DFT algorithms using dynamic programming

Abstract: A broad class of efficient discrete Fourier transform algorithms is developed by partitioning short DFT algorithms into factors. The factored short DFT's are combined into longer DFT's using multi-dimensional index maps. By exploiting a property which allows some of the factors to commute, a large set of possible DFT algorithms is generated which contains both the prime factor algorithm (PFA) and the Winograd Fourier transform algorithm (WFTA) as special cases. The problem of finding an algorithm from this class which is optimal with respect to the specific add, multiply, and data transfer characteristics of a particular implementation is posed, and a highly effective dynamic programming algorithm is presented as a solution.

Numerical evaluation of the radiation from unbaffled, finite plates using the FFT

Abstract: An iteration technique is described which numerically evaluates the acoustic pressure and velocity on and near unbaffled, finite, thin plates vibrating in air. The technique is based on Rayleigh’s integral formula and its inverse. These formulas are written in their angular spectrum form so that the fast Fourier transform (FFT) algorithm may be used to evaluate them. As an example of the technique the pressure on the surface of a vibrating, unbaffled disk is computed and shown to be in excellent agreement with the exact solution using oblate spheroidal functions. Furthermore, the computed velocity field outside the disk shows the well‐known singularity at the rim of the disk. The radiated fields from unbaffled flat sources of any geometry with prescribed surface velocity may be evaluated using this technique. The use of the FFT to perform the integrations in Rayleigh’s formulas provides a great savings in computation time compared with standard integration algorithms, especially when an array processor can be used to implement the FFT.

Method of performing real input fast fourier transforms simultaneously on two data streams

Abstract: Simultaneous FFT's are calculated for two real input sequences utilizing a two channel recursive FFT structure. The method of operating such a structure disclosed includes the log 2 N stages (for an N-point transform) known in the prior art plus a final unscrambling stage. A method of arranging the outputs of the log 2 N-th stage for use as inputs to the unscrambling stage is disclosed which results in substantially unscrambled FFT outputs for the two sequences without the need for extra unscrambling hardware.

An algorithm for the numerical evaluation of the Hankel and Abel transforms

Abstract: A procedure for the efficient numerical evaluation of the Hankel and Abel transforms is proposed. The Abel transform is reduced to a convolution which is evaluated in part analytically and in part with an FFT. The Hankel transform is obtained by following the Abel transform with an FFT.

Fast Fourier Transform Algorithms with Applications

Abstract: After introduction of the discrete Fourier transform and a short description of its main properties we concentrate on a discussion of some of the various methods which have been used for the derivation of fast Fourier transform (FFT) algorithms. The paper closes with applications of FFT procedures in the numerical calculation of Fourier coefficients, fast multiplication of large integers and computations which involve circulant matrices.

Architecture for two dimensional fast fourier transform

Abstract: A novel architecture and circuitry for implementing a new fast fourier transform algorithm which does not require a very large core memory and also does not require a transpose of a matrix. A pipelined and parallel architecture implements the two dimensional fast fourier transform on an array of input data values, with the transformation being performed by a plurality of serially arranged pass stages. Each pass stage includes an input shuffle arrangement for receiving an ordered set of input data from a row or column of a two dimensional matrix of such input data values, and for performing a shuffle operation thereon to produce a shuffled order of the input data. Each pass stage further includes a plurality of identical switching circuits coupled in parallel to receive the shuffled order of input data. Each switching circuit includes an arithmetic logic unit which receives four input data values and performs four data transformations thereon to produce four output data values, with each of the four data transformations including a first operation of selective addition or subtraction of the four input data values, followed by a second operation of selective multiplication by an exponential multiplier.

New algorithms for the multidimensional discrete Fourier transform

Abstract: We exhibit new algorithms for DFT(p; k), the discrete Fourier transform on a k-dimensional data set with p points along each array, where p is a prime. At a cost of additions only, these algorithms compute DFT(p; k) with (pk- 1)/(p - 1) distinct DFT(p; 1) computations.

Application of the one-dimensional Fourier transform for tracking moving objects in noisy environments

Abstract: In Riddle and Rajala (1981), an algorithm was presented which operates on an image sequence to identify all sets of pixels having the same velocity. The algorithm operates by performing a transformation in which all pixels with the same two-dimensional velocity map to a peak in a transform space. The transform can be decomposed into applications of the one-dimensional Fourier transform and therefore can gain from the computational advantages of the FFT. The aim of this paper is the concern with the fundamental limitations of that algorithm, particularly as relates to its sensitivity to image-disturbing parameters as noise, jitter, and clutter. A modification to the algorithm is then proposed which increases its robustness in the presence of these disturbances.

Monolithic fast fourier transform circuit

Abstract: A fast Fourier transform circuit formed on a single chip, including a fast multiplier-accumulator circuit which, in the preferred embodiment, employs a modified form of Booth's algorithm, an adder circuit, a read-only memory for storing FFT twiddle factors, and a random access memory for holding a set of input complex quantities and for receiving intermediate and final results in an in-place FFT operation. In the preferred embodiment, the FFT twiddle factors are stored in Booth's code for greater speed of operation. Control and timing circuitry on the same chip generates control signals and address codes in order to perform a sequence of butterfly computations by repeated use of the multiplier-accumulator and adder circuits, to generate FFT coefficients in the random access memory.

Two-dimensional modeling of an optical disk readout.

Abstract: The design of high density digital optical storage systems requires a thorough understanding of the various interactions affecting the integrity of both data and servo signals under various readout conditions. Here, a 2-D Fourier optic model of the readout system is presented along with an efficient calculational procedure based on 2-D fast Fourier transforms (FFT). The model is utilized to examine the effects of data density, focus errors, tracking errors, lens fill conditions, phase pit depth, etc. on the quality of the readout signal. The derivation of servo signals is also studied through the modeling of a simple diffraction based scheme. It is shown that a reasonably constant servo signal (small variations with focus errors, track depth, or added amplitude data) can be generated by this technique provided that a λ/8 phase groove is present. The same technique can be shown to be unacceptable if an amplitude or λ/4 phase groove is utilized.

On the errors involved in computing the empirical characteristic function

Abstract: This paper considers discretisation errors involved in using the Fast Fourier Transform to compute the empirical characteristic function efficiently. A simple improvement to the usual histogram discretisation scheme is shown to reduce the mean square error considerably, as the grid size tends to zero. Simulation results show that the improvement is just as good in practical cases. The theoretical results are applied to the efficient calculation of kernel density estimates, described in Silverman (1982).

Performance Analysis of FFT Algorithms on Multiprocessor Systems

Abstract: A decimation-in-time radix-2 fast Fourier transform (FFT) algorithm is considered here for implementation in multiprocessors with shared bus, multistage interconnection network (MIN), and in mesh connected computers. Results are derived for data allocation, interprocessor communication, approximate computation time, and speedup of an N point FFT on any P available processing elements (PE's). Further generalization is obtained for a radix-r FFT algorithm. An N X N point two-dimensional discrete Fourier transform (DFT) implementation is also considered when one or more rows of the input data matrix are allocated to each PE.