Showing papers on "Discrete-time Fourier transform published in 1981"

Estimation of Periods from Unequally Spaced Observations

Abstract: A better estimation of the power spectrum of a time series formed with unequally spaced observations may be obtained by means of a data-compensated discrete Fourier transform. This transform is defined so as to include the uneven spacing of the dates of observation and weighting of the corresponding data. The accurate determination of the peak heights allows one to design harmonic filters and thus to make a more certain choice among peaks of similar height and also to discriminate peaks that are just aliases of other peaks. The theory is applied to simulated time series and also to true observational data.

Noise in Fourier self-deconvolution.

Abstract: A general formula for computing changes in the signal-to-noise ratio of a spectrum resulting from the Fourier self-deconvolution procedure is derived. Self-deconvolution reduces the intrinsic halfwidths of lines by a factor K, which is in practice limited by the noise in the spectrum. With the help of the derived formula, the rate of decrease in the SNR as a function of K for eight different smoothing (apodization) functions is studied. With high K values there are significant differences in the SNR as a result of the use of different smoothing functions. With K = 4 difference of more than 1 order of magnitude between two extreme cases is demonstrated, and with K = 5 a difference of almost 2 orders of magnitude in the SNR is predicted.

Linear systems, Fourier transforms, and optics

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On Some Fourier Methods for Inference

Abstract: Common statistical procedures such as maximum likelihood and M-estimation admit generalized representations in the Fourier domain. The Fourier domain provides fertile ground for approaching a number of difficult problems in inference. In particular, the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well. A numerical illustration involving the symmetric stable laws is given.

The Phase Retrieval Problem

Abstract: A method for solving the phase retrieval problem is proposed. The method consists of measuring the modulus of the Fourier transform of the unknown function and the modulus of the Fourier transform of the product of the unknown function and an exponential. From these two measurements, the location of the complex zeros of the analytic continuation of the Fourier transform of the unknown function may easily and quickly be deduced and the unknown function constructed.

Short-time Fourier analysis of sampled speech

Abstract: The theoretical basis for the representation of a speech signal by its short-time Fourier transform is developed. A time-frequency representation for linear time-varying systems is applied to the speech-production model to formulate a quasi-stationary representation for the speech waveform. Short-time Fourier analysis of the resulting representation yields the relationship between the short-time Fourier transform of the speech and the speech-production model.

Frequency shift in Fourier analysis

Abstract: Error analysis of the frequency determination by conventional Fourier method was performed. Taking into consideration only equidistant data, estimates were obtained on the accuracy of the frequencies deduced by analysing a noise-free signal containing not more than two sinusoidal components with closely spaced frequencies. An important consequence of these results was the empirical distribution function of the frequencies for signals contaminated by noise and containing sufficiently distant frequencies.

Area-Time Optimal VLSI Networks for Computing Integer Multiplications and Discrete Fourier Transform

Abstract: In this paper we present a class of VLSI networks for computing the Discrete Fourier Transform and the product of two N-bit integers. These networks match, within a constant factor, the known theoretical lower-bound O(N2) to the area × (time)2 measure of complexity. While this paper's contribution is mainly theoretical, it points toward very practical directions: we show how to design multipliers with area A = O(N) and time T=O(√N) on one hand, and A=0((N/log2N)2), T = O(log2N) on the other. Both of these designs should be contrasted with the currently available multipliers, whose performances are A=O(N), T=O(N) or even A=O(N2), T=O(N).

Limited-Angle Three-Dimensional Reconstructions Using Fourier Transform Iterations And Radon Transform Iterations

Abstract: The principles of limited-angle reconstruction of space-limited objects using the concepts of "allowed cone" and "missing cone" in Fourier space are discussed. The distortion of a point source resulting from setting the Fourier components in the missing cone to zero has been calculated mathematically, and its bearing on the convergence of an iteration scheme involving Fourier transforms has been analyzed in detail. It was found that the convergence rate is fairly insensitive to the position of the point source within the boundary of the object, apart from an edge effect which tends to enhance some parts of the boundary in reconstructing the object. Another iteration scheme involving Radon transforms was introduced and compared to the Fourier transform method in such areas as root mean square error, stability with respect to noise, and computer reconstruction time.

Reflector antenna fields--An exact aperture-like approach

Abstract: A new computational approach is presented which allows fast analysis of radiation from large reflector antennas. For an aperture a Fourier transform (FT) relationship does exist between far-field and aperture distribution. Accordingly, the far field can be exactly reconstructed from the knowledge of approximately one sample per lobe (Shannon-Whittaker theorem applied at Nyquist rate). The finite reflector curvature introduces an extra factor in the radiation integral so that the radiation integral is no longer a FT. In order to overcome this difficulty a new pseudosampling expansion, which explicitly takes into account the extra factor, is developed. For parabolic reflector the sampling functions are related to the Fresnel integrals, and the far field can be exactly reconstructed in terms of aperture far-field samples, which can be computed using the fast Fourier transform (FFT). Numerical computations and error analysis show the excellent performance of the method, which can be generalized to deal with arbitrary reflector surfaces and near-field evaluation.

Method for continuing Fourier spectra given by the fast Fourier transform

Abstract: A new method for continuing the Fourier spectrum of data of finite extent has been developed. Almost perfect restoration was obtained for noise-free data. Much improved resolution was obtained for error-laden data. Many forms of peak data lend themselves easily to the application of additional constraints that further enhance resolution almost to the theoretical limit.

Non-linear wave propagation solutions by Fourier transform perturbation

Abstract: A Fourier transform perturbation method is developed and used to obtain uniformly valid asymptotic approximations of the solution of a class of one-dimensional second order wave equations with small non-linearities. Multiple time scales are used and the initial-value problem on the infinite line is solved by Fourier transforming the wave equation and expanding the Fourier transform in powers of the small parameter. The non-linearity involves only the first partial derivatives of the dependent variable and the determination of the leading approximation is reduced to the solution of a pair of coupled non-linear ordinary differential equations in Fourier space. Examples are given involving a convolution non-linearity and a Van-der-Pol non-linearity.

Fourier transform signal processor

Abstract: A hybrid opto-electronic method and apparatus for Fourier transform measurements of temporal signals are disclosed. The magnitude and phase of the signal transform are system outputs. High resolution spectral analysis operation is accomplished by proper spatial phase corrections, equivalent to aperture synthesis. Configurations for one-dimensional and two-dimensional input data formats are described. The disclosed method and apparatus can process signals with continuous spectral content as well as narrow spectral lines.

Continuation of discrete Fourier spectra using a minimum-negativity constraint

Abstract: A method for continuing Fourier spectra applicable to spectra given by the discrete Fourier transform is presented. From the principle of minimizing the sum of the square of the negative values of the restored function, a set of simultaneous nonlinear equations is obtained. The only means of solution at present is iterative, but computational time is comparable with that for noniterative methods. Excellent restoration is obtained for the sharply attenuated spectrum of deconvolved infared peaks. The numerical procedure developed here lends itself easily to the inclusion of additional constraints to enhance resolution further. The constraints of minimum negativity and finite extent may both be enforced together on pertinent data with only slight modification of the procedure.

Fast computation of Fourier transforms at arbitrary frequencies

Abstract: An algorithm is proposed for computing the Fourier Transform (FT) of a uniformly sampled signal at arbitrary frequencies. In most of the applications, the algorithm retains the computational efficiency of the Fast Fourier Transform (FFT) algorithm. The method is based on the fact that the FT at an arbitrary frequency can be expressed as a weighted sum of its Discrete Fourier Transform (DFT) coefficients. In the proposed method, these weights are suitably approximated so that the desired FT is very nearly the sum of (i) a few dominant terms of the sum of the DFT which are computed directly, and (ii) the DFT of a new sequence obtained by multiplying the original sequence with a sawtooth function. The number of directly computed terms is so chosen that the error of approximation does not exceed the specified limits. The computational aspects of the algorithm and its error behavior with typical signals have been critically examined.

Spectral Decomposition of EEG Intervals Using Walsh and Fourier Transforms

Abstract: This paper presents a comparison of the use of features derived from Walsh and Fourier transforms for classification of short segments of EEG data. The result was that features obtained from Fourier transforms produced consistently better classification results than Walsh function-derived features.

Walsh Versus Fourier Estimators of the EEG Power Spectrum

Abstract: Walsh-based discrete algorithms offer a fast, simple alternative to Fourier methods for estimating EEG power spectra, but their performance has been criticized. Previous comparisons of Fourier and Walsh estimators are reexamined, and they are compared experimentally, on simulated EEG. The evidence suggests that they should perform comparably for EEG monitoring.

A DFT-based residual-excited linear predictive coder (RELP) for 4.8 and 9.6kb/s

Abstract: The proposed RELP system is based on a DFT (discrete Fourier transform) of the linear prediction residual. The base-band Fourier coefficients, derived from frequencies 0 to 800 Hz, are transmitted to the receiver. For encoding, the Fourier coefficients are converted to polar representation reiφ. The quantization accuracy of both magnitude and phase angle is controlled by an adaptive-bit assignment algorithm. At the receiver, the non-transmitted Fourier coefficients are replaced by base-band coefficients which have been frequency-shifted pitch-synchronously. In the 4.8 kb/s version, the spectral translation algorithm is controlled only by the transmitted pitch frequency. In the 9.6 kb/s version, however, additional information for reconstruction of the harmonic structure in the high-frequency region is computed at the transmitter. Audio tapes of both versions are presented.

A new prime factor FFT algorithm

Abstract: This paper presents an approach to calculating the discrete Fourier transform (DFT) using a prime factor algorithm (PFA). A very simple indexing scheme is employed that results in a flexible, modular program that very efficiently calculates the DFT in-place. A modification of this indexing scheme gives a new algorithm with the output both in-place and in-order at a slight cost in flexibility. This means only 2N data storage is needed for a length N complex FFT and no unscrambling is necessary. The basic part of a FORTRAN program is given. A speed comparison shows the new algorithm to be faster than both the Cooley-Tukey and the nested Winograd algorithms.

Positioning device for picture

Abstract: PURPOSE:To perform high-speed positioning without the intervention of operator's judgement, by finding the shift extent of every divided partial area, and performing position shift compensation according to a compensation function calculated from found shift extents. CONSTITUTION:In an Y-directional sum register 4a and an X-directional sum register 4b, data on partial areas to be processed among picture data read out of a picture memory 1 are set respectively. Then, DFT (discrete Fourier transform) calculating circuits 5a and 5b calculate Fourier transform values of those picture data. The Fourier transform value of the other picture data to be superimposed is also calculated to calculate an X-directional and an Y-directional movement extent from the found Fourier transform value, and they are added to center coordinates set in corresponding partial part addresses of registers 3a and 3b. A minimum square-low calculating circuit 7 obtain the coefficient of the polynomial of a compensation function T for obtaining data in registers 3a and 3b.