Showing papers on "Discrete-time Fourier transform published in 1983"

Interpolation Algorithms for Discrete Fourier Transforms of Weighted Signals

Abstract: A new scheme is presented for the determination of the parameters that characterize a multifrequency signal. The essential innovation is that the signal is weighted before the discrete Fourier transform (DFT) is calculated from which the frequencies and complex amplitudes of the various components of the signal are obtained by interpolation. It is shown that by using the Hanning window for tapering substantial improvements are achieved in the following respects: i) more accurate results are obtained for interpolated frequencies, etc., ii) harmonic interference is much less troublesome even if many tones with comparable strengths are present in the spectrum, iii) nonperiodic signals can be handled without an a priori knowledge of the tone frequencies. The stability of the new method with respect to noise and arithmetic roundoff errors is carefully examined.

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.

Signals and Systems: Continuous and Discrete

Abstract: 1. Signal and System Modeling Concepts. 2. System Modeling and Analysis in the Time Domain. 3. The Fourier Series. 4. The Fourier Transform and Its Applications. 5. The Laplace Transformation. 6. Applications of the Laplace Transform. 7. State-Variable Techniques. 8. Discrete-Time Signals and Systems. 9. Analysis and Design of Digital Filters. 10. The Discrete Fourier Transform and Fast Fourier Transform Algorithms. Appendix A: Comments and Hints on Using MATLAB. Appendix B: Functions of a Complex Variable--Summary of Important Definitions and Theorems. Appendix C: Matrix Algebra. Appendix D: Analog Filters. Appendix E: Mathematical Tables. Appendix F: Answers to Selected Problems. Appendix G: Index of MATLAB Functions Used. Index.

Redundancy, the Discrete Fourier Transform, and Impulse Noise Cancellation

Abstract: The relationship between the discrete Fourier transform and error-control codes is examined. Under certain conditions we show that discrete-time sequences carry redundant information which then allow for the detection and correction of errors. An application of this technique to impulse noise cancellation for pulse amplitude modulation transmission is described.

Applications of Discrete and Continuous Fourier Analysis

Abstract: An applications oriented, introductory text covering the concepts and properties of Fourier Analysis. Emphasizes applications to real scientific and engineering problems. Defines the Fourier series, Fourier transform, and discrete Fourier transform. Includes over 200 illustrations.

Signal reconstruction from signed Fourier transform magnitude

Abstract: In this paper, we show that a one-dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed Fourier transform magnitude (magnitude and 1 bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multidimensional sequence from its Fourier transform magnitude. Reconstruction examples obtained using this algorithm are also provided.

Some aspects of band-limited signal extrapolation: Models, discrete approximations, and noise

Abstract: We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise η is bounded by a known number, max \eta(x)| leq \epsilon . This convergence is obtained by using normal families of entire functions in ¢nand some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

Superresolution of Fourier transform spectroscopy data by the maximum entropy method.

Abstract: The maximum entropy method (MEM) is applied to the interferogram data obtained using the technique of Fourier transform spectroscopy for estimating its spectrum with a resolution far exceeding the value set by the spectrometer. For emission line data, the MEM process is directly used with the interferogram data in place of the regular Fourier transformation process required in Fourier transform spectroscopy. It produces a spectral estimate with an enhanced resolution. For absorption data with a broad background spectrum, the method is applied to a modified interferogram which corresponds to the Fourier transform of the absorptance spectrum. Two results are presented to demonstrate the power of the technique: for the visible emission spectrum of a spectral, calibration lamp and for the infrared chloroform absorption spectrum. Included in the paper is a discussion of the problems associated with practical use of the MEM.

Stability of unique Fourier-transform phase reconstruction

Abstract: The problem of Fourier-transform phase reconstruction from the Fourier-transform magnitude of multidimensional discrete signals is considered. It is well known that, if a discrete finite-extent n-dimensional signal (n ≥ 2) has an irreducible z transform, then the signal is uniquely determined from the magnitude of its Fourier transform. It is also known that this irreducibility condition holds for all multidimensional signals except for a set of signals that has measure zero. We show that this uniqueness condition is stable in the sense that it is not sensitive to noise. Specifically, it is proved that the set of signals whose z transform is reducible is contained in the zero set of a certain multidimensional polynomial. Several important conclusions can be drawn from this characterization, and, in particular, the zero-measure property is obtained as a simple byproduct.

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.

The finite fourier transform of a stationary process

Abstract: Publisher Summary The Fourier transform has proved of substantial use in most fields of science. It has proved of special use to statisticians concerned with stationary process data or concerned with the analysis of linear time-invariant systems. This chapter describes some of the uses and properties of Fourier transforms of stochastic processes. The Fourier transform turns up in the problems of functional approximation and interpolation. In seismic engineering, the Fourier transforms of observed strong motion records are taken as design inputs and corresponding responses of structures evaluated prior to construction. There are various classes of functions that may be viewed as subject to a harmonic analysis. Quite a different class of functions is provided by the realizations of stationary stochastic processes. Fourier transforms at distinct frequencies and based on nonintersecting data stretches may be approximated by independent normals. The variance of the approximating normal is proportional to the power spectrum of the series.

Effects of additive noise on signal reconstruction from Fourier transform phase

Abstract: The effects of additive noise in the given phase on signal reconstruction from the Fourier transform phase are experimentally studied. Specifically, the effects on the sequence reconstruction of different methods of sampling the degraded phase of the number of nonzero points in the sequence, and of the noise level, are examined. A sampling method that significantly reduces the error in the reconstructed sequence is obtained, and the error is found to increase as the number of nonzero points in the sequence increases and as the noise level increases. In addition, an averaging technique is developed which reduces the effects of noise when the continuous phase function is known. Finally, as an illustration of how the results in this paper may be applied in practice, Fourier transform signal coding is considered. Coding only the Fourier transform phase and reconstructing the signal from the coded phase is found to be considerably less efficient (i.e., a higher bit rate is required for the same mean-square error) than reconstructing from both the coded phase and magnitude.

The Talbot Effect as a Sequence of Quadratic Phase Corrections of the Object Fourier Transform

Abstract: A new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed. All properties of the self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform. The angular dimensions of the self-images, as well as the notions of the constant of periodical field configuration and the self-image vergence, are introduced. They allow the characterization, in a uniform manner, of the field distribution in the whole space independently of the chosen self-image plane. The equivalency between the self-imaging phenomenon and the image defocusing by an optical system are considered. The general formulae for the harmonics analysis of the intensity distribution are derived.

A note on two weight function conditions for a Fourier transform norm inequality

Abstract: Recent papers have given two different conditions on pairs of nonnegative weight functions that insure that a Fourier transform norm inequality holds in R". With additional assumptions these conditions were also shown to be implied by the norm inequality. A direct proof is given here that these conditions are equivalent; this can be used to simplify some of the proofs in those papers.

Recursive discrete Fourier transformation with unevenly spaced data

Abstract: New recursive techniques for Fourier spectral analysis are reported, for which ongoing spectral estimates are generated from unevenly spaced data in real time. The algorithms are robust and computationally efficient, and are well suited to state variable form involving real number calculations. These methods are particularly attractive in filtering and signal processing applications where signals are not necessarily sampled at a uniform rate.

Comments on "The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform"

Abstract: When one imposes a nonnegativity constraint, one usually can reconstruct a two-dimensional sequence of finite support from the modulus of its Fourier transform using an iterative algorithm, even when file initial estimate is an array of random numbers.

The Fourier Transform of the Characteristic Function of a Set, Vanishing on AN Interval

Abstract: A set E with 0 < meas E < +∞ is constructed for which the Fourier transform of its characteristic function vanishes on an interval. The set is the union of a sequence of intervals whose lengths can be estimated asymptotically above and below. The construction is based on an infinite-dimensional version of the implicit function theorem. Bibiography: 6 titles.

Two VLSI Structures for the Discrete Fourier Transform

Abstract: Two VLSI structures for the computation of the discrete Fourier transform are presented. The first structure is a pipeline working concurrently on different transforms. It is shown that it matches, within a constant factor, the theoretical lower bounds for area versus data rate. The second structure is a simple modification of the first one; it works on a single transform at a time, and it matches within a constant factor the theoretical area-time lower bounds.

A Real Formalism Of Discrete Fourier Transform In Terms Of Skew-Circular Correlations And Its Computation By Fast Correlation Techniques

Abstract: Discrete fourier transform is represented as a real transform through using number groups and removing redundancy. The resulting configuration is further written in terms of (skew) circular correlations, which can be implemented by fast correlation techniques. The number of data points considered is a power of 2, even though the method can be generalized to any number of data points.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Magnitude-only reconstruction of two-dimensional sequences with finite regions of support

Abstract: In this paper, a conceptual algorithm for reconstructing a two-dimensional (2-D) complex-valued finite sequence from an adequate set of samples of the magnitude of its Fourier transform is presented. In particular, one obtains, at least theoretically, all solutions of the 2-D magnitude-only reconstruction problem, provided that the modulus of the DFT is available in a sufficiently large set of points. However, the practicability of this algorithm is limited to sequences with relatively small regions of support. The key for developing the method is shown to be an appropriate mapping of 2-D finite sequences into 1-D ones, such that 2-D discrete correlation can be formulated in terms of ordinary 1-D discrete correlation.

Signal Reconstruction From Fourier Transform Amplitude

Abstract: In this paper, we show that a one-dimensional or multi-dimensional sequence is uniquely specified under mild restrictions by its Fourier transform amplitude (magnitude and one bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multi-dimensional sequence from its Fourier transform amplitude. Reconstruction examples obtained using this algorithm are also provided.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Fourier analyses: A mathematical and geometric explanation

Abstract: Fourier analyses are used in electrophysiological research to reduce EEG data to an interpretable, analyzable form. This article outlines the mathematical similarities and differences between Fourier transforms and fast Fourier transforms. A geometric explanation of the application of fast Fourier transforms and a Fourier series to theta-band EEG data is also included in this article.

Switched-capacitor realization of a discrete Fourier transformer

Abstract: Frequency sampling filter approach is described to compute the discrete Fourier transform (DFT). The resulting configuration requires delay elements and differential summers which are shown to be realizable by simple stray-insensitive switched-capacitor (SC) circuits. The proposed scheme finds applications where short data blocks are processed like in radar.

Correlations, contrasts, and components: Fourier analysis in a more familiar terminology

Abstract: The Fourier transform partitions the energy in a waveform into the sum of the energies of simpler components. This process is the same as the partitioning of variance into linear contrasts and is a way of measuring the correlation between the waveform and each member of a family of prototype model waveforms. Such a partitioning will often, but not always, result in a meaningful decomposition of the original waveform.

Conversion factors from Walsh coefficients to Fourier coefficients

Abstract: We have proposed a computational algorithm for the discrete Fourier transform (DFT) via the discrete Walsh transform (DWT). However, the calculation equations for the conversion factors from the DWT coefficients to the DFT coefficients have not been shown. This paper presents the equations for the conversion factors.

Fast discrete cosine transform algorithm for systolic arrays

Abstract: A fast algorithm for an N-point discrete cosine transform (DCT) is derived from a 4N-point Winograd Fourier transform algorithm (WFTA). This algorithm, which has the same form as Winograd's Fourier transform and convolution algorithms, is suitable for a high-speed implementation using one-bit systolic arrays.

Measurement of Small Secondary Flow Components by Photon Correlation Velocimetry

Abstract: Cross-beam laser velocimetry using Bragg shift and photon correlation has been utilized to measure a weak secondary flow component in the presence of strong primary flow at right angles to it. To obtain this velocity component with adequate accuracy, the apparent Doppler frequency was determined by Fourier transformation of the correlogram followed by two steps of interpolating correction. The remaining frequency error may be as little as 10-2 j f where integral multiples of j f are the abscissa values for which the discrete Fourier transform is given. Finally a method is described which allows precise orientation of the cross-beam fringes at right angles to the secondary flow component; it makes use of symmetry properties of the flow in question.