Showing papers on "Non-uniform discrete Fourier transform published in 1982"
TL;DR: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT) based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT.
Abstract: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.
243 citations
TL;DR: In this article, the Laplace transform of the solution for TEM soundings over an N-layer earth was derived and used to invert it numerically using the Gaver-Stehfest algorithm.
Abstract: Calculations for the transient electromagnetic (TEM) method are commonly performed by using a discrete Fourier transform method to invert the appropriate transform of the solution. We derive the Laplace transform of the solution for TEM soundings over an N-layer earth and show how to use the Gaver-Stehfest algorithm to invert it numerically. This is considerably more stable and computationally efficient than inversion using the discrete Fourier transform.
149 citations
90 citations
01 May 1982
TL;DR: This paper presents various conditions that are sufficient for reconstructing a discrete-time signal from samples of its short-time Fourier transform magnitude, for applications such as speech processing.
Abstract: This paper presents various conditions that are sufficient for reconstructing a discrete-time signal from samples of its short-time Fourier transform magnitude. For applications such as speech processing, these conditions place very mild restrictions on the signal as well as the analysis window of the transform. Examples of such reconstruction for speech signals are included in the paper.
79 citations
TL;DR: In this article, a steplike waveform is converted into a duration-limited one which preserves the spectrum of the original waveform and is suitable for discrete Fourier transform (DFT) computations.
Abstract: A steplike waveform which has attained its final value is converted into a duration-limited one which preserves the spectrum of the original waveform and is suitable for discrete Fourier transform (DFT) computations. The method, which is based upon the response of a time-invariant linear system excited by a rectangular pulse of suitable duration, is first applied to continuous waveforms and then to discrete (sampled) waveforms. The difference (errors) between the spectra of a continuous waveform and a discrete representation of it are reviewed.
74 citations
54 citations
01 Jan 1982
TL;DR: In this article, the main properties of the discrete Fourier transform (DFT) are summarized and various fast DFT computation techniques known collectively as the Fast Fourier Transform (FFT) algorithm are presented.
Abstract: The object of this chapter is to briefly summarize the main properties of the discrete Fourier transform (DFT) and to present various fast DFT computation techniques known collectively as the fast Fourier transform (FFT) algorithm. The DFT plays a key role in physics because it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. The use of DFT analysis methods has increased dramatically since the introduction of the FFT in 1965 because the FFT algorithm decreases by several orders of magnitude the number of arithmetic operations required for DFT computations. It has thereby provided a practical solution to many problems that otherwise would have been intractable.
50 citations
TL;DR: In this article, the authors evaluated the relationship derived in terms of the continuous Fourier integral transform with the discrete Fourier transform for potential field geophysical studies and showed that the discrete transform can be made essentially equivalent to the integral transform if, before sampling, the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi...
Abstract: In the application of harmonic analysis to potential‐field geophysical studies, relationships derived in terms of the continuous Fourier integral transform are evaluated in terms of the discrete Fourier transform. The discrete transform, obtained by transforming a finite number of equispaced samples of the actual aperiodic continuous function, is too low at the dc level and increasingly too high in the high frequencies, compared with the theoretical integral transform. As a consequence, overly restrictive limitations must be placed on high‐frequency‐amplifying operators such as differentiation and downward continuation. Also, a spurious and troublesome azimuthal distortion occurs in the discrete Fourier analysis of three‐dimensional (3-D) (map) data represented as grids. The discrete transform can be made essentially equivalent to the integral transform if, before sampling, the continuous aperiodic input function is made periodic by shifting the function by integer multiples of the data interval and summi...
48 citations
TL;DR: In this article, the Fourier transform was used to analyze the self-potential anomaly due to a two-dimensional inclined sheets of finite depth extent using the frequency domain using Fourier Transform.
Abstract: The self-potential anomaly due to a two-dimensional inclined sheets of finite depth extent has been analysed in the frequency domain using the Fourier transform. Expression for the Fourier amplitude and phase spectra are derived. The Fourier amplitude and phase spectra are analysed so as to evaluate the parameters of the sheet. Application of this method on two anomalies (synthetic and field data) has given good results.
46 citations
TL;DR: In this paper, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform, which are applicable to both one-dimensional and multi-dimensional sequences.
Abstract: Recently, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform. These conditions are distinctly different from the minimum or maximum phase requirement and are applicable to both one-dimensional and multi-dimensional sequences. Under the specified conditions, several numerical algorithms have been developed to reconstruct a sequence from its phase. In this paper, we review the recent theoretical results pertaining to the phase-only reconstruction problem, and we discuss in detail two iterative numerical algorithms for performing the reconstrucction.
34 citations
TL;DR: A modified version of Burrus' prime factor fast Fourier transform program is described, which implements the in-place, in-order algorithm for variable transform sizes.
Abstract: This paper describes a modified version of Burrus' prime factor fast Fourier transform program. The modifications produce a general-purpose program which implements the in-place, in-order algorithm for variable transform sizes. Speed tests show the resulting program to be faster than a program using a separate reordering pass.
TL;DR: In this article, an algorithm is presented which takes advantage of the fact that minimization of the error term can be accomplished by minimizing the distance between the origin of the polar coordinate system in the calculation of the Fourier series and the shape centroid.
Abstract: The ability to test for similarities and differences among families of shapes by closed-form Fourier expansion is greatly enhanced by the concept of homology. Underlying this concept is the assumption that each term of a Fourier series, when compared to the same term in another series, represents the “same thing”. A method that ensures homology is one which minimizes the “centering error,” as reflected in the first harmonic term of the Fourier expansion. The problem is to chose a set of edge points derived from a much larger, but variable, number of edge points such that a valid homologous Fourier series can be calculated. Methods are reviewed and criteria given to define a “proper” solution. An algorithm is presented which takes advantage of the fact that minimization of the “error term” can be accomplished by minimizing the distance between the origin of the polar coordinate system in the calculation of the Fourier series and the shape centroid. The use of this algorithm has produced higher quality solutions for quartz grain provenance studies.
TL;DR: A review of the discrete Fourier transform, emphasizing the use of DFT in direct and indirect methods of time domain signal processing and a number of applications to communications.
Abstract: A review of the discrete Fourier transform, emphasizing the use of DFT in direct and indirect methods of time domain signal processing. T HE discrete Fourier transform (DFT), implemented as a computationally efficient algorithm called the fast Fourier transform (FFT), has found application to all aspects of signal processing. These applications include time domain processing as well as frequency domain processing. The proper noun \"Fourier\" may elicit images of frequency domain data and, by these images, restrict the vista of applications of this important tool.. W e will review the DFT with emphasis on perspectives which facilitate time domain processing. In particular, we will review a number of applications to communications.
TL;DR: Cumulative probability distributions that occur in radar and sonar detection problems are calculated directly from the characteristic function by using a Fourier series, a valuable tool in system performance studies.
Abstract: Cumulative probability distributions that occur in radar and sonar detection problems are calculated directly from the characteristic function by using a Fourier series. The error in the result is controlled by two parameters which can be adjusted to suit the application. The technique is applied to the problem of determining the detection performance of consecutive discrete Fourier transforms (DFTs) for a narrowband Gaussian signal with a rectangular spectrum. Since the characteristic function is used directly in its product form this technique does not suffer from the numerical problems associated with the partial fraction approach. The technique can handle many different problems in a single computational structure making it a valuable tool in system performance studies.
TL;DR: A new algorithm for the calculation of the Fourier transform of sampled time functions is described, based on second‐degree polynomial interpolations between the sample points, which was found to be significantly more accurate than the conventionally used discrete Fouriertransform (DFT).
Abstract: A new algorithm for the calculation of the Fourier transform of sampled time functions is described. The algorithm is especially applicable to the Fourier analysis of nonperiodic signals which are not band limited. The method is based on second‐degree polynomial interpolations between the sample points. The obtained continuous approximation of the signal allows the determination of the Fourier transform analytically. In the case of exponentially decaying functions the algorithm was found to be significantly more accurate than the conventionally used discrete Fourier transform (DFT). The computing time is only about twice the time required by the fast Fourier transform (FFT) algorithm.
TL;DR: A procedure is given for the computation of the transient state occupancy probabilities of the M/M/1 queue, which makes use of the inverse discrete Fourier transform computed by means of the fast Fouriertransform.
Abstract: A procedure is given for the computation of the transient state occupancy probabilities of the M/M/1 queue. The method makes use of the inverse discrete Fourier transform, computed by means of the fast Fourier transform. It avoids the direct evaluation of modified Bessel functions and sidesteps difficulties due to the computation of very large and very small intermediate quantities.
TL;DR: A generalized running discrete transform with respect to arbitrary transform bases is introduced, and the generalized transform to the running discrete Fourier z and short-time discrete Fouriers transforms is related.
Abstract: This paper introduces a generalized running discrete transform with respect to arbitrary transform bases, and relates the generalized transform to the running discrete Fourier z and short-time discrete Fourier transforms. Concepts associated with the running and short-time discrete Fourier transforms such as 1) filter bank implementation, 2) synthesis of the original sequence by summation of the filter bank outputs, 3) frequency sampling, and 4) recursive implementations are all extended to the generalized transform case. A formula is obtained for computing the transform coefficients of a segment of data at time n recursively from the transform coefficients of the segment of data at time n - 1. The computational efficiency of this formula is studied, and the class of transforms requiring the minimum possible number of arithmetic operations per coefficient is described.
01 May 1982
TL;DR: In this paper a 2400 bit/s implementation of the DFT-vocoder is discussed and the harmonic-sieve technique for pitch extraction combines very well with this scheme because it is based on hopping-DFT as well.
Abstract: The DFT-vocoder is based on speech analysis and synthesis using the discrete Fourier transform (DFT). Analysis is done using hopping-DFT and spectral parameters are obtained by a piece-wise constant approximation of the amplitude spectrum. The harmonic-sieve technique for pitch extraction combines very well with this scheme because it is based on hopping-DFT as well. Synthesis is achieved by convolution of the generated excitation signal with the inverse-DFT of the reconstructed piece-wise constant amplitude spectrum. In this paper a 2400 bit/s implementation of the DFT-vocoder is discussed.
TL;DR: In this paper, two original methods are presented for deconvolving such transforms for signals containing significant noise, and the results of numerical experiments with noisy data are presented in order to demonstrate the capabilities and limitations of the methods.
Abstract: Time series or spatial series of measurements taken with nonuniform spacings have failed to yield fully to analysis using the Discrete Fourier Transform (DFT). This is due to the fact that the formal DFT is the convolution of the transform of the signal with the transform of the nonuniform spacings. Two original methods are presented for deconvolving such transforms for signals containing significant noise. The first method solves a set of linear equations relating the observed data to values defined at uniform grid points, and then obtains the desired transform as the DFT of the uniform interpolates. The second method solves a set of linear equations relating the real and imaginary components of the formal DFT directly to those of the desired transform. The results of numerical experiments with noisy data are presented in order to demonstrate the capabilities and limitations of the methods.
TL;DR: A simple derivation of Glassman's general N fast Fourier transform, and corresponding FORTRAN program, is presented, based upon a representation of the discrete Fourier Transform matrix as a product of sparse matrices.
Abstract: : A simple derivation of Glassman's general N fast Fourier transform, and corresponding FORTRAN program, is presented. This fast Fourier transform is based upon a representation of the discrete Fourier transform matrix as a product of sparse matrices. (Author)
TL;DR: A third possibility is suggested: a direct Fourier transform which takes advantage of certain properties of a spike train and works much faster than a common Fouriertransform.
Abstract: A spike train may be represented by a superposition of Dirac delta-functions. One of the simplest ways of converting such a comb function into a continuous function is to use a Fourier transform. In general there are two possibilities, both of which have their disadvantages: the direct transform which is extremely time-consuming, and the fast Fourier transform of the low pass filtered comb function; the latter method, although quicker, often requires a greater storage capacity than is readily available. In the present paper, therefore, a third possibility is suggested. Essentially, it is a direct Fourier transform which takes advantage of certain properties of a spike train. The corresponding algorithm works much faster than a common Fourier transform.
29 Mar 1982
TL;DR: There are many different methods in use to scrample voice signals, but newer equipment, which is realized by digital circuitry, allow us to use both methods, band splitting and time division, at the same time.
Abstract: There are many different methods in use to scrample voice signals Two of them seem to be of special importance: band-splitting and time-division In existing devices for scrambling analog signals often only on of these methods is implemented However, newer equipment, which is realized by digital circuitry, allow us to use both methods, band splitting and time division, at the same time
03 May 1982
TL;DR: It is shown that, given the boundary values, phase retrieval becomes a linear problem and the interior points of the two-dimensional sequence may easily be deduced from the boundaries values and the magnitude of its Fourier transform or, equivalently, the autocorrelation function of the sequence.
Abstract: In this paper, the importance of the boundary values of a two dimensional sequence in the phase retrieval problem is investigated. Specifically, it is shown that, given the boundary values, phase retrieval becomes a linear problem. Therefore, the interior points of the two-dimensional sequence may easily be deduced from the boundary values of the sequence and the magnitude of its Fourier transform or, equivalently, the autocorrelation function of the sequence. Furthermore, although the determination of the boundary values from only Fourier transform magnitude information is, in general, a non-trivial problem, it is shown that the shape of the region of support of the two-dimensional sequence determines the ease with which the boundary values may be determined.
TL;DR: In this article, a s-spline fit is made to the input function, and the Fourier transform of the set of B-splines is performed analytically for a possibly nonuniform mesh.
Abstract: Finite Fourier integrals of functions possessing jumps in value, in the first or in the second derivative, are shown to be evaluated more efficiently, and more accurately, using a continuous Fourier transform (CFT) method than the discrete transform method used by the fast Fourier transform (FFT) algorithm. A s-spline fit is made to the input function, and the Fourier transform of the set of B-splines is performed analytically for a possibly nonuniform mesh. Several applications of the CFT method are made to compare its performance with the FFT method. The use of a 256-point FFT yields errors of order 10~2, whereas the same information used by the CFT algorithm yields errors of order 10~7—the machine accuracy available in single precision. Comparable accuracy is obtainable from the FFT over the limited original domain if more than 20,000 points are used.
01 May 1982
TL;DR: A sampling method is developed to significantly reduce the error in the reconstructed sequence, and the error is found to increase as the number of non-zero points in the sequence increases and as the noise level increases.
Abstract: The effects of noise in the given phase on signal reconstruction from the Fourier transform phase are studied. Specifically, the effects of different methods of sampling the degraded phase, of the number of non-zero points in the sequence, and of the noise level on the sequence reconstruction are examined. A sampling method is developed to significantly reduce the error in the reconstructed sequence, and the error is found to increase as the number of non-zero 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.
TL;DR: This paper reports on the application of a new FFT algorithm, first described by Winograd, to the calculation of diffraction OTF that yields the same accuracy as that obtained by the Cooley-Tukey method but is up to four times faster.
Abstract: Although fast Fourier transform (FFT) algorithms based on the Cooley-Tukey method have been widely used for the computation of optical transfer function (OTF), the need for yet faster algorithms remains. This is particularly so since desk-top computers with modest speed and memory size have become essential tools in optical design. In this paper we report on the application of a new FFT algorithm, first described by Winograd, to the calculation of diffraction OTF. The algorithm is compared both in speed and in accuracy with the commonly used radix-2 FFT and with an autocorrelation method employing the Gaussian quadrature integration technique. It is found that the new algorithm yields the same accuracy as that obtained by the Cooley-Tukey method but is up to four times faster. Some other advantages and drawbacks are discussed.
TL;DR: The sectionalized Fourier Transform has many applications in time-domain signal processing using modern array digital computers and has proved advantageous in underwater acoustic applications.
Abstract: The sectionalized Fourier transform of a band-limited signal (defined as a Fourier transform which is computed over incremented temporal sections of the function) is equivalent to basebanding, filtering, and sampling the signal in time domain. Spectral windowing is employed, through appropriately summing a sequence of the Fourier transform bins, to control the passband and leakage characteristics of the resulting filter. This in turn controls the distortion of the signal induced as a result of the transform process. The use of the sectionalized Fourier transform is exploited to conveniently and rapidly map the cross-correlation envelope of narrow-band signals over the time-register Doppler-ratio (ambiguity) plane. By using the ambiguity kernel \exp(i2\pi\alphaft) as an approximation of signal time compression (or expansion), the coherence between transformed signals (along the Doppler-ratio axis) may further be expedited through use of the discrete Fourier transform. The resulting error is negligible when the time-bandwidth product of the process is less than the inverse of the maximum Doppler ratio employed. The resulting algorithms have proved advantageous in underwater acoustic applications. It is concluded that the sectionalized Fourier Transform has many applications in time-domain signal processing using modern array digital computers.