Showing papers on "Fast Fourier transform published in 1979"

High-Accuracy Analog Measurements via Interpolated FFT

Abstract: By use of an interpolated fast-Fourier-transform (FFT) algorithms are developed for multiparameter measurements upon periodic signals. Eight pertinent measurements, such as fundamental frequency, phase, and amplitude, are made with enhanced accuracy compared to existing algorithms, including tapered-window-FFT algorithms. For the more general case of nonharmonic multitone signals also the method is shown to yield exact amplitudes and phases if the tone frequencies are known beforehand. These measurements are useful in a variety of applications ranging from analog testing of printed-circuit boards to measurement of Doppler signals in radar detection.

A Sinusoidal Family of Unitary Transforms

Abstract: A new family of unitary transforms is introduced. It is shown that the well-known discrete Fourier, cosine, sine, and the Karhunen-Loeve (KL) (for first-order stationary Markov processes) transforms are members of this family. All the member transforms of this family are sinusoidal sequences that are asymptotically equivalent. For finite-length data, these transforms provide different approximations to the KL transform of the said data. From the theory of these transforms some well-known facts about orthogonal transforms are easily explained and some widely misunderstood concepts are brought to light. For example, the near-optimal behavior of the even discrete cosine transform to the KL transform of first-order Markov processes is explained and, at the same time, it is shown that this transform is not always such a good (or near-optimal) approximation to the above-mentioned KL transform. It is also shown that each member of the sinusoidal family is the KL transform of a unique, first-order, non-stationary (in general), Markov process. Asymptotic equivalence and other interesting properties of these transforms can be studied by analyzing the underlying Markov processes.

Fast hankel transforms

Abstract: Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type: Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*. If the input function f(λ exp (iω)) is known to be analytic in the region o < λ < ∞, |ω|≤ω0 of the complex plane, we can show that the absolute error on the output function is less than (K(ω0)/r) · exp (−ρω0/Δ), Δ being the logarthmic sampling distance. Due to the explicit expansions of H* the tails of the infinite summation ((m−n)Δ) can be handled analytically. Since the only restriction on the order is ν > − 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).

Very Fast Fourier Transform Algorithms Hardware for Implementation

Abstract: A new method of deriving very fast Fourier transform (FFT) algorithms is described. The resulting algorithms do not employ multiplication and have a form suitable for high performance hardware implementations. The complexity of the algorithms compares favorably to the recent results of Winograd [1].

A plane-polar approach for far field reconstruction from near-field measurements

Abstract: It is well-known that the far field of an arbitrary antenna may be calculated from near-field measurements. Among various possible nearfield scan geometries, the planar configuration has attracted considerable attention. In the past the planar configuration has been used with a probe scanning a rectangular geometry in the near field, and computation of the far field has been made with a two-dimensional fast Fourier transform (FFT). The applicability of the planar configuration with a probe scanning a polar geometry is investigated. The measurement process is represented as a convolution derivable from the reciprocity theorem. The concept of probe compensation as a deconvolution is then discussed with numerical results presented to verify the accuracy of the method. The far field is constructed using the Jacobi-Bessel series expansion and its utility relative to the FFT in polar geometry is examined. Finally, the far-field pattern of the Viking high gain antenna is constructed from the plane-polar near-field measured data and compared with the previously measured far-field pattern. Some unique mechanical and electrical advantages of the plane-polar configuration for determining the far-field pattern of large and gravitationally sensitive space antennas are discussed. The time convention exp ( j \omega r ) is used but is suppressed in the formulations.

Cochlear macromechanics: Time domain solutions

Abstract: In this paper we report on a new method of solving a previous derived, two‐dimensional model, integral equation for basilar membrane (BM) motion. The method uses a recursive algorithm for the solution of an initial‐value problem in the time domain, combined with a fast Fourier transform (FFT) convolution in the space domain at each time step. Thus, the method capitalizes on the high speed and accuracy of the FFT yet allows the BM to have nonlinear mechanical properties. Using the new method we compute (linear) solutions for various choices of model parameters and compare the results to the experimental measurements of Rhode. [J. Acoust. Soc. Am. 49, 1218–1231 (1971)]. We also demonstrate the effect of including longitudinal stiffness along the BM and conclude that it is useful in matching the high‐frequency slope as measured by Rhode.

Computing the Fast Fourier Transform on a vector computer

Abstract: Two algorithms are presented for performing a Fast Fourier Transform on a vector computer and are compared on the Control Data Corporation STAR-100. The relative merits of the two algorithms are shown to depend upon whether only a few or many independent transforms are desired. A theorem is proved which shows that a set of independent transforms can be computed by performing a partial transformation on a single vector. The results of this theorem also apply to nonvector machines and have reduced the average time per transform by a factor of two on the CDC 6600 computer.

Negation can be exponentially powerful

Abstract: Among the most remarkable algorithms in algebra are Strassen's algorithm for the multiplication of matrices and the Fast Fourier Transform method for the convolution of vectors. For both of these problems the definition suggests an obvious algorithm that uses just the monotone operations + and ×. Schnorr [18] has shown that these algorithms, which use t(n3) and T(n2) operations respectively, are essentially optimal among algorithms that use only these monotone operations. By using subtraction as an additional operation and exploiting cancellations of computed terms in a very intricate way Strassen showed that a faster algorithm requiring only O(n2.81) operations is possible. The FFT method for convolution achieves O(nlog n) complexity in a similar fashion. The question arises as to whether we can expect even greater gains in computational efficiency by such judicious use of cancellations. In this paper we give a positive answer to this, by exhibiting a problem for which an exponential speedup can be attained using {+,−,×} rather than just {+,×} as operations. The problem in question is the multivariate polynomial associated with perfect matchings in planar graphs. For this a fast algorithm is implicit in the Pfaffian technique of Fisher and Kasteleyn [6,8]. The main result we provide here is the exponential lower bound in the monotone case.

Fast computation of discrete Fourier transforms using polynomial transforms

Abstract: Polynomial transforms, defined in rings of polynomials, have been introduced recently and have been shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFT's and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA). We also describe new split nesting and split prime factor techniques for computing large DFT's from a small set of short DFT's with a minimum number of operations.

Algorithms: Their Complexity and Efficiency

Abstract: Evaluation of Polynomials Iterative Processes Direct Methods for Solving Sets of Linear Equations The Fast Fourier Transform Fast Multiplications of Numbers Internal Sorting External Sorting Searching.

Implementation of FFT Structures Using the Residue Number System

Abstract: This paper considers the implementation of a fast Fourier transform (FFT) structure using arrays of read-only memories. The arithmetic operations are based entirely on the residue number system. The most important aspect of the structure relates to the scaling arrays, which are required to prevent overflow. Because of the limitations of the number system, scaling factors have to be chosen on an a priori basis. This paper develops optimum procedures for choosing both scaling factors and the position of scaling arrays in the structure. Some examples are presented relating to the filtering of speech via a convolutional filter structure.

Simplified fast fourier transform butterfly arithmetic unit

Abstract: An improved organization for a FFT analyzer (or periodic function analyzer) having a reduced computing complexity. A modified organization comprises a simplified butterfly arithmetic unit in which the usual two coefficient registers or memories are required. By utilizing the registers as sources of a respective sum of and difference between sets of phase-shifted cosine values, the mechanization of the complex multiplier for such arithmetic butterfly unit in microcircuit or "chip" form may be further simplified to two controllable accumlators controlled by an exclusive-NOR gate logic system responsive to the states of the complex sampled inputs of a sampled signal epoch of interest. In this way, a more efficient and higher speed device is provided for the multiplication of complex variables.

Nondestructive determination of refractive index profile of an optical fiber: fast Fourier transform method

Abstract: A nondestructive technique of determining the refractive index profile of an optical fiber is presented. This method involves collecting the pathlength data of rays passing through the fiber due to side-illumination and taking the fast Fourier transform of these data followed by a numerical integration. It is found that this technique yields results in good agreement with measurements from the near-field scanning technique. The advantage of the present method is that it can readily be extended to fibers of noncircular cross section.

Digital Simulation of Nonlinear Random Waves

Abstract: Digital realizations of unidirectional nonlinear random seas correct to second order in an ocean of finite depth are simulated from three types of two-parameter theoretical spectra and are compared with measured hurricane-generated realizations by means of chi-square goodness-of-fit measures computed from the Gram-Charlier probability distribution in which the statistical measures of skewness and the excess of kurtosis are determined from the measured hurricane-generated realizations. The finite Fourier transform (FFT) algorithm is shown to be an efficient method for nonlinear simulations since the FFT coefficients are complex and, therefore, capable of retaining the nonlinear random phase interactions. The second-order nonlinear simulations demonstrate improved third-order and fourth-order statistical moments compared to the linear Gaussian simulations. Previous comparisons with measured wave forces on cylindrical pilings have demonstrated improvements in the statistics of random wave force predictions computed by digital filter methods as a result of these improved nonlinear random sea simulations.

Fast FFT-based algorithm for phase estimation in speckle imaging

Abstract: This paper describes a fast direct algorithm for obtaining least-squares phase estimates from arrays of noisy phase differences. The algorithm uses the fast Fourier transform to diagonalize and decouple the system of equations which results from the application of the least-squares criterion. It is accurate and stable, and is perhaps an order of magnitude faster than the best iterative method. The effectiveness of the algorithm has been demonstrated by using it in connection with the Knox–Thompson speckle-imaging procedure to restore an optical object perturbed by simulated atmospheric turbulence. Representative results are discussed in the paper.

A simple fixed-point error bound for the fast Fourier transform

Abstract: Error bounds for the computation of the fast Fourier transform in fixed-point arithmetic are derived for any arithmetic number base and for any prime factorization of the data array length. The intended application is for signal processing with minicomputers. Errors arising from inaccurate sine coefficients and from limited arithmetic precision are considered. The arithmetic error depends essentially on shifts of the data array that may be required to avoid overflow of the computer word. Our closest bound requires knowledge of where shifts occur and is best computed in parallel with the Fourier transform. For the case that such program modification is not feasible, we derive an error bound for a posteriori calculation and an a priori error estimate. Our bounds are for the maximum error because little is gained at the expense of considerably greater complexity for probabilistic error bounds.

Short-time Fourier analysis tecniques for FIR system identification and power spectrum estimation

Abstract: A wide variety of methods have been proposed for system modeling and identification. To date, the most successful of these methods have been time domain procedures such as least squares analysis, or linear prediction (ARMA models). Although spectral techniques have been proposed for spectral estimation and system identification, the resulting spectral and system estimates have always been strongly affected by the analysis window (biased estimates), thereby reducing the potential applications of this class of techniques. In this paper we propose a novel short-time Fourier transform analysis technique in which the influences of the window on a spectral estimate can essentially be removed entirely (an unbiased estimator) by linearly combining biased estimates. As a result, section (FFT) lengths for analysis can be made as small as possible, thereby increasing the speed of the algorithm without sacrificing accuracy. The proposed algorithm has the important property that as the number of samples used in the estimate increases, the solution quickly approaches the least squares (theoretically optimum) solution. The method also uses a fixed Fourier transform length independent of the amount of data being analyzed, allowing the estimate to be recursively updated as more data is made available. The method assumes that the system is a finite impulse response (FIR) system.

Properties of the Multidimensional Generalized Discrete Fourier Transform

Abstract: In this work the generalized discrete Fourier transform (GFT), which includes the DFT as a particular case, is considered. Two pairs of fast algorithms for evaluating a multidimensional GFT are given (T-algorithm, F-algorithm, and T′-algorithm, F′-algorithm). It is shown that in the case of the DFT of a vector, the T-algorithm represents a form of the classical FFT algorithm based on a decimation in time, and the F-algorithm represents a form of the classical FFT algorithm based on decimation in frequency. Moreover, it is shown that the T′-algorithm and the T-algorithm involve exactly the same arithmetic operations on the same data. The same property holds for the F′-algorithm and the F-algorithm. The relevance of such algorithms is discussed, and it is shown that the T′-algorithm and the F′-algorithm are particularly advantageous for evaluating the DFT of large sets of data.

Efficient FFT Simulation of Digital Time Sequences

Abstract: A stacked inverse finite Fourier transform (FFT) algorithm is presented that will efficiently synthesize a discrete random time sequence of N values from only N/2 complex values having a desired known spectral representation. This stacked inverse FFT algorithm is compatible with the synthesis of discrete random time sequences that are used with the more desirable periodic-random type of dynamic testing systems used to compute complex-valued transfer functions by the frequency-sweep method. An application to the generation of large random surface gravity waves by a hinged wavemaker in a large-scale wave flume demonstrates excellent agreement between the desired theoretical spectral representation and the smoothed, measured spectral representation for two types of two-parameter theoretical spectra as a result of the lengthier realization made possible by the stacked FFT algorithm.

High-Order Fast Elliptic Equation Solvers

Abstract: An algorithm for approximating certain classes of elliptic partial differential equations on a rectangle is presented. The algorithm uses high-order 9-point difference approximations to the Helmholtz-type (fourth-order) or Poisson (sixth-order) equations and the fast Fourier transform. Compared to efficient second-order fast direct, methods for smooth problems, the execution time is reduced by a large factor, typically 50 for the Helmholtz-type equations and over 100 for the Poisson problem. Comparisons with two high-order fast direct methods indicate the superiority of the algorithm.

On the use of symmetry in FFT computation

Abstract: It is well known that if a finite duration, N-point sequence x(n) possesses certain symmetries, the computation of its discrete Fourier transform (DFT) can be obtained from an FFT of size N/2 or smaller. This is accomplished by first preprocessing the sequence, taking the FFT of the processed sequence, and then postprocessing the results to give the desired transform. In this paper we show how a similar approach can be used for sequences which are known to have only odd harmonics. The approach is shown to be essentially the dual of the known method for time symmetry. Computer programs are included for implementing the special procedures discussed in this paper.

Fast algorithms for the conjugate periodic function

Abstract: Two fast algorithms for the approximate computation of the conjugate periodic function are described They are based on the fast Fourier transform and enable us to reduce the expenses toO (N logN) operations compared withO (N 2) operations for Wittich's classical method The second algorithm, for which an ALGOL 60 procedure is listed, allows to evaluate the conjugate function on the even (or odd) numbered lattice points separately (This feature is important for some applications)

Some unique signal processing applications in power system planning

Abstract: Three applications of digital signal processing in power system planning are discussed. The use of a vector radix 2-D FFT in a spatial load growth model has reduced computation time for spatial convolutions by an order of magnitude. The character of a power system is used to derive a set of 2-D digital filters that represent the power system's sensitivity to spatial load forecast-design errors. System design filters provide a useful way of analyzing long range power system needs.