Showing papers on "Discrete-time Fourier transform published in 1976"
TL;DR: A method based on the Fourier convolution theorem is developed for the analysis of data composed of random noise, plus an unknown constant "base line," plus a sum of (or an integral over a continuous spectrum of) exponential decay functions.
531 citations
TL;DR: In this article, a new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced, and three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose.
Abstract: A new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced. From this class, three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose.
291 citations
TL;DR: This paper discusses a digital formulation of the phase vocoder, an analysis-synthesis system providing a parametric representation of a speech waveform by its short-time Fourier transform, designed to be an identity system in the absence of any parameter modifications.
Abstract: This paper discusses a digital formulation of the phase vocoder, an analysis-synthesis system providing a parametric representation of a speech waveform by its short-time Fourier transform. Such a system is of interest both for data-rate reduction and for manipulating basic speech parameters. The system is designed to be an identity system in the absence of any parameter modifications. Computational efficiency is achieved by employing the fast Fourier transform (FFT) algorithm to perform the bulk of the computation in both the analysis and synthesis procedures, thereby making the formulation attractive for implementation on a minicomputer.
240 citations
TL;DR: In this paper, an alternative form of the fast Fourier transform (FFT) is developed, which has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary.
Abstract: An alternative form of the fast Fourier transform (FFT) is developed. The new algorithm has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary. The advantages of the new form would, therefore, seem to be most pronounced in systems for which multiplication are most costly.
161 citations
TL;DR: This correspondence shows that the amount of work can be cut to doing two single length FFT's, which is equivalent to doing one double length fast Fourier transform.
Abstract: Ahmed has shown that a discrete cosine transform can be implemented by doing one double length fast Fourier transform (FFT). In this correspondence, we show that the amount of work can be cut to doing two single length FFT's.
77 citations
70 citations
TL;DR: One-dimensional and two-dimensional generalized discrete Fourier transforms (GFTs) are introduced in this article, and the result holds also for the DFT, as it is a particular case of the GFT.
Abstract: One-dimensional and two-dimensional generalized discrete Fourier transforms (GFT) are introduced. If a one-dimensional vector A is fractured into a two-dimensional matrix B, a one-dimensional GFT on A and a two-dimensional GFT on B give the same result and require the same number of operations to be computed. The result holds also for the DFT, as it is a particular case of the GFT.
47 citations
TL;DR: Some of the important features of the fast Fourier transform which are relevant to its increasing application to biomedical data are reviewed and a distinction is made between the power spectrum of ergodic signals, computed from the autocorrelation function, and the frequency spectrum of nonstationary biomedical signals.
Abstract: The fast Fourier transform (f.f.t.) is a powerful technique which facilitates analysis of signals in the frequency domain. This paper reviews some of the important features of the fast Fourier transform which are relevant to its increasing application to biomedical data. A distinction is made between the power spectrum of ergodic signals, computed from the autocorrelation function, and the frequency spectrum of nonstationary biomedical signals. The major practical pitfalls that are encountered in applying the f.f.t. technique to biomedical data are discussed, and practical hints for avoiding such pitfalls are suggested.
44 citations
TL;DR: In this article, the spectrum of a magnetic or a gravity anomaly due to a body of a given shape with either homogeneous magnetization or uniform density distribution can be expressed as a product of the Fourier transforms of the source geometry and the Green's function.
Abstract: The spectrum of a magnetic or a gravity anomaly due to a body of a given shape with either homogeneous magnetization or uniform density distribution can be expressed as a product of the Fourier transforms of the source geometry and the Green's function. The transform of the source geometry for any irregularly-shaped body can be accurately determined by representing the body as closely as possible by a number of prismatic bodies. The Green's function is not dependent upon the source geometry. So the analytical expression for its transform remains the same for all causative bodies. It is, therefore, not difficult to obtain the spectrum of an anomaly by multiplying the transform of the source geometry by that of the Green's function. Then the inverse of this spectrum, which yields the anomaly in the space domain, is calculated by using the Fast Fourier Transform algorithm. Many examples show the reliability and accuracy of the method for calculating potential field anomalies.
42 citations
01 Mar 1976
TL;DR: In this paper, an efficient algorithm for the DFT of N point symmetric real-valued series with time samples taken as odd multiples of half the sampling period T/2 and frequency samples are taken as 1/2NT is presented.
Abstract: An efficient algorithm is obtained for the DFT of N point symmetric real-valued series if time samples are taken as odd multiples of half the sampling period T/2 and frequency samples are taken as odd multiples of 1/2NT.
TL;DR: In this article, the linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by noise.
Abstract: Summary The linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by ' noise '. When ' noise ' in the data is of concern this method achieves a maximum decrease in the variance of the Fourier transform estimate for a minimum sacrifice in resolution, thereby optimizing the trade-off between resolution and accuracy. The effects of data gaps are easily treated and it is shown that it may sometimes be desirable to interpolate these gaps even though a large variance must be ascribed to the fabricated data. We also apply the Backus-Gilbert technique to the calculation of the reverse Fourier transform, and an application to the downward continuation of potential field data is given.
TL;DR: In this paper, the Fourier transform noise was examined experimentally using specially prepared programs, and it was found that the dynamic range observable following a Fourier transformation is proportional to the computer's word length and inversely proportional to both the number of transformed words and the memory locations full at the outset of the transform.
TL;DR: Number theoretic transforms that can be used for the convolution of complex integer sequences are defined and a unified setting is provided wherein these transforms may be defined for all odd moduli, thus extending recent results on this topic.
Abstract: Number theoretic transforms (NTT's) that can be used for the convolution of complex integer sequences are defined. A unified setting is provided wherein these transforms may be defined for all odd moduli, thus extending recent results on this topic. Multiplication-free implementation of certain of these transforms is possible. When these transforms are used to implement convolution, the resulting computation is exact.
Patent•
19 Jul 1976TL;DR: In this paper, an automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals is presented, where the initial calculation as well as the equalisation proper are conducted entirely within the frequency domain.
Abstract: An automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals. The initial calculation as well as the equalization proper are conducted entirely within the frequency domain. Overlapping moving window samplings are employed together with the discrete Fourier transformation and a sparse inverse discrete Fourier transformation to provide the equalized time domain output signals.
TL;DR: The problem of evaluating successively the discrete Fourier transform on ordered sets of N elements staggered of M is considered, and three procedures for solving such a problem are given, of which two are recursive and one nonrecursive.
Abstract: In this work the problem of evaluating successively the discrete Fourier transform (DFT) on ordered sets of N elements staggered of M is considered. Three procedures for solving such a problem are given, of which two are recursive and one nonrecursive. The complexity of each procedure, in number of complex multiplications, is about (N/2) \log_{2} 4M .
TL;DR: Techniques for dealing with signals having a high dynamic range in Fourier transform nmr are discussed, and the limitations imposed by the transform itself are pointed out.
TL;DR: In this paper, a formula to determine the characteristic function of N x N matrix by discrete Fourier series is given, which is based on the Fourier transform of the matrix.
Abstract: A formula to determine the characteristic function of N x N matrix by discrete Fourier series is given.
01 Jan 1976
TL;DR: In this article, the Fourier transform of a radial integrable function on Euclidean space has (1 − l)/2 derivatives, which is a stronger version of Theorem A incase p = 1.
Abstract: generalization of the Fourier transform of a radial integrable function onEuclidean «-space) has [(« - l)/2] derivatives; in [6] Schwartz has improvedSchoenberg's result, actually obtaining a stronger version of Theorem A incase p = 1. Our main result, Theorem 1, represents a combination of theresults of Schwartz and Tomas in case of ordinary derivatives; the extensionto fractional ones is new. Furthermore, Theorem 1 allows us to deducenecessary conditions for radial Fourier Mpq multipliers (in an elementary way).A comparison of these conditions with sufficient ones enlightens to someextent the structure of radial Fourier multipliers.The author is obliged to the referee for pointing out the papers ofSchoenberg [5] and Schwartz [6].The following notation will be used: y, z E R", s, ..., x E R; S is the set
Patent•
01 Mar 1976
TL;DR: In this article, an arrangement for computing the discrete Fourier transform intended for converting N samples of a real signal in the time domain to N real Fourier coefficients is presented. But this device is implemented with a conventional Fourier transformer of the order N/4, to which an input computer unit and an output computer unit are connected in which a small number of multiplications of complex numbers is performed.
Abstract: An arrangement for computing the discrete Fourier transform intended for converting N samples of a real signal in the time domain to N real Fourier coefficients. This device is implemented with a conventional Fourier transformer of the order N/4, to which an input computer unit and an output computer unit are connected in which a small number of multiplications of complex numbers is performed.
TL;DR: The amended line spread function notation is employed in the piecewise isoplanatic approximation treatment of variant systems as well as in developing a sampling theorem for space-variant system^.^ Expressions of this form employing the conventional line spreadfunction notation of Eq.
Abstract: YMBOLS OMEMCLATURE tion is improved terminology: technical expression rather than technical content. They will be refereed in the same manner as Letters, but by different criteria. In the process of characterizing a linear system, the system line spread function (impulse response) is normally written in the f ~ r m l-~ where S [-1 is the system operator, 6(x) is the Dirac delta, and x, and xi are, respectively, the output and input variables. Upon assumption of space-invariance (isoplanicity), one writes That is, the line spread function shifts directly with the input impulse and thus depends only on the coordinate difference X o-4'. At least five author^^-^ have utilized the less widely used line spread function notation The most. obvious advantage of this notation, as noted by Lohmann and P a r i ~ , ~ is the cleaner transition to the invariant case. The line spread function merely becomes independent of its second argument: Note that the function h(x,-[) here is equivalent to that in Eq. (2). A second advantage of the amended line spread function notation is the straightforward manner in which one can express a space-variant system's transfer function. Consider first the procedure in the isoplanatic case. One probes the input with a single-shifted impulse, 6(xi-,$), finds the corresponding output, h (x ,-t), shifts this output to obtain h(x,), and last performs a Fourier transform to arrive at the system transfer function. One may perform an analogous procedure using the amended line spread function notation to arrive at the transfer function of a space-variant system: where f, is the spatial frequency domain variable and 3,, [ ] is the Fourier transform operator in x, defined by Space-variant transfer functions of the form of Eq. (5) are employed in the piecewise isoplanatic approximation treatment of variant systems6 as well as in developing a sampling theorem for space-variant system^.^ Expressions of this form employing the conventional line spread function notation of Eq. (1) are less intuitive. Other difficulties arising when using the conventional notation are discussed by Kail~ith.~ The amended line spread function lends itself nicely to Fourier transformation with respect to its variation variable [. Such a computation, for example, is necessary in applications of the space-variant system sampling t h e ~ r e r n. ~ One looks at If this expression is band limited in u for all x,. (i.e., if it has a finite …
Patent•
19 Jul 1976TL;DR: In this paper, an automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals is presented, where the initial calculation as well as the equalisation proper are conducted entirely within the frequency domain.
Abstract: An automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals The initial calculation as well as the equalization proper are conducted entirely within the frequency domain Overlapping moving window samplings are employed together with the discrete Fourier transformation and a sparse inverse discrete Fourier transformation to provide the equalized time domain output signals
TL;DR: In this article, the authors examined the impact of finite register lengths on data acquisition, computation of the fast Fourier transform (FFT), and post-FFT spectral manipulations, and concluded that the minimum recommended register length is 27 bits.
Abstract: Finite registers used in computations act as additional noise sources in infrared Fourier transform spectroscopy The relationship between these noise sources and classical noise sources is examined The impact of finite register lengths on data acquisition, computation of the fast Fourier transform (FFT), and post-FFT spectral manipulations leads to the conclusion that the minimum recommended register length is 27 bits
TL;DR: An algorithm is developed for making magnetic field “reduction-to-the-pole” computations using two-dimensional Fourier series using a “look-up table” to reduce the number of trigonometric functions to be evaluated.
TL;DR: The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept.
Abstract: Truncated series expansion is used to obtain discrete-time windows which are optimal in the least-square sense for a given number of terms. The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept. As a consequence, the resulting windows are easy to obtain and economical to implement in practical applications.
12 Apr 1976
TL;DR: The filtering method and its implementation are shown to compare favorably with DFT and FFT filtering and the comparison is presented in terms of complexity of hardware implementation and error accumulation.
Abstract: A discrete filtering technique based on the discrete Hilbert Transform (DHT) is presented. The filtering method and its implementation are shown to compare favorably with DFT and FFT filtering. The comparison is presented in terms of complexity of hardware implementation and error accumulation. The DHT itself and derived discrete filter transforms are each presented in matrix form and may be implemented with the same hardware in either series or parallel form. In this manner a large variety of filtering characteristics may be implemented by changes of the stored coefficients only. While the filtering process is a circular convolution, direct convolution filters are obtainable also, by using one row of the matrix only, and are realizable in the form of an FIR (non-recursive) digital filter.
TL;DR: In this paper, the Laplace transform is approximated at exponentially spaced samples and analysis frequencies, and the ratio of the intervals between pairs of adjacent sampling positions is a constant greater than one.
Abstract: An estimate of the spectrum is based on the Laplace transform which is approximated at exponentially spaced samples and analysis frequencies. In this approximation the ratio of the intervals between pairs of adjacent sampling positions is a constant greater than one. The choice of this constant is influenced by the desired analysis bandwidth and by sampling effects. If analysis frequencies are spaced the same as sampling positions, this approximation becomes a discrete correlation. which can be computed by a fast Fourier transform (FFT) or a number theoretic transform. Except at low-analysis frequencies, the analysis bandwidth is "constant-Q," i.e., it is proportional to the analysis frequency. With a white noise input the noise in the computed spectrum is roughly constant at each analysis frequency. The numbers of samples and computations required for exponential spacing of samples and frequencies can be less than those required for equidistant spacing. Better performance at some (but not all) analysis frequencies is provided by a two-sided sampling arrangement consisting of a juxtaposition of the basic one-sided sampling arrangement and its mirror image.