Showing papers on "Discrete-time Fourier transform published in 1990"
TL;DR: In this paper, the decay properties of the Fourier transform of a compactly supported function can be used to define non-quasianalytic weight functions with respect to continuous and continuous derivatives.
Abstract: Classes of non-quasianalytic functions are classically defined by imposing growth conditions on the derivatives of the functions. It was Beuding [1] (see Bjorck [2]) who pointed out that decay properties of the Fourier transform of a compactly supported function can be used for this purpose equally well. In the present article we modify Beuding's approach. More precisely, we call w: [0,00[--+ [0, oo[ a weight function if w is continuous and satisfies
295 citations
TL;DR: In this article, the authors propose to use multifractal analysis in reciprocal space as a tool to characterise, in a statistical and global sense, the nature of the Fourier transform of geometrical models for atomic structures.
Abstract: The authors propose to use multifractal analysis in reciprocal space as a tool to characterise, in a statistical and global sense, the nature of the Fourier transform of geometrical models for atomic structures. This approach is especially adequate for shedding some new light on a class of structures introduced recently, which exhibit 'singular scattering'. Using the language of measure theory, the Fourier intensity of these models is presumably singular continuous, and therefore represents an intermediate type of order, between periodic or quasiperiodic structures, characterised by Bragg peaks (atomic Fourier transform), and amorphous structures, which exhibit diffuse scattering (absolutely continuous Fourier transform). This general approach is illustrated in several examples of self-similar one-dimensional sequences and structures, generated by substitutions. A special emphasis is put on the relationship between the nature of the Fourier intensity of these models and the f( alpha ) spectrum obtained by multifractal analysis in reciprocal space.
113 citations
04 Dec 1990
TL;DR: In this article, a 2D Fourier transform technique involving both spatial and time transformations is presented to identify and measure the amplitudes of individual Lamb wave modes, and the results obtained from both numerical and experimental investigations of Lamb waves propagating in steel plates are presented using an isometric projection.
Abstract: The key problem associated with the quantitative measurement of the characteristics of propagating Lamb waves is that more than one wave mode can exist at any given frequency. Therefore, a simple Fourier transformation from the time to the frequency domain cannot distinguish between the different modes. A 2-D Fourier transform technique involving both spatial and time transformations from which the required information can be obtained is presented. The results obtained from both numerical and experimental investigations of Lamb waves propagating in steel plates are presented using an isometric projection, which gives a 3-D view of the wave-number dispersion curves. The results show the effectiveness of using the 2-D Fourier transform method to identify and measure the amplitudes of individual Lamb modes. >
98 citations
Book•
01 Jan 1990TL;DR: In this paper, a classification of continuous-time systems is presented, based on the Fourier Transform and its relation to the Gibbs Phenomenon, and the Laplace Transform.
Abstract: 1. Representing Signals. Continuous-Time vs. Discrete-Time Signals. Periodic vs. Aperiodic Signals. Energy and Power Signals. Transformations of the Independent Variable. Elementary Signals. Other Types of Signals. 2. Continuous - Time Systems. Classification of Continuous-Time Systems. Linear Time- Invariant Systems. Properties of Linear Time-Invariant Systems. Systems Described by Differential Equations. State Variable Representations. 3. Fourier Series. Orthogonal Representations of Signals. The Exponential Fourier Series. Dirichlet conditions. Properties of the Fourier Series. Systems with Periodic Inputs. The Gibbs Phenomenon. 4. The Fourier Transform. The Continuous-Time Fourier Transform. Properties of the Fourier Transform. Applications of the Fourier Transform. Duration-Bandwidth Relationships. 5. The Laplace Transform. The Bilateral Laplace Transform. The Unilateral Laplace Transform. Bilateral Transforms Using Unilateral Transforms. Properties of the Unilateral Laplace Transform. The Inverse Laplace Transform. Simulation Diagrams for Continuous-Time Systems. Applications of the Laplace Transform. State Equations and the Laplace Transform. Stability in the s Domain. 6. Discrete-Time Systems. Elementary Discrete-Time Signals. Discrete-Time Systems. Periodic Convolution. Difference-Equation Representation of Discrete-Time Systems. Stability of Discrete Time Systems. 7. Fourier Analysis of Discrete-Time Systems. Fourier-Series Representation of Discrete-Time Periodic Signals. The Discrete-Time Fourier Transform. Properties of the Discrete-Time Fourier Transform. Fourier Transform of Sampled Continuous-Time Signals. 8. The Z-Transform. The Z-Transform. Convergence of the Z-Transform. Properties of the Z-Transform. The Inverse Z-Transform. Z-Transfer Functions of Casual Discrete-Time Systems. Z-Transform Analysis of State-Variable Systems. Relation Between the Z-Transform and the Laplace Transform. 9. The Discrete Fourier Transform. The Discrete Fourier Transform and Its Inverse. Properties of the DFT. Linear Convolution Using the DFT. Fast Fourier Transforms. Spectral Estimation of Analog Signals Using the DFT. 10. Design of Analog and Digital Filters. Frequency Transformations. Design of Analog Filters. Digital Filters. Appendices.
80 citations
03 Apr 1990
TL;DR: Circular convolution-multiplication relationships for the discrete cosine transform (DCT) that are similar to those forThe discrete Fourier transform (DFT) are developed and can be used to filter an image in the frequency domain as an approximation of circular convolution in the spatial domain.
Abstract: Circular convolution-multiplication relationships for the discrete cosine transform (DCT) that are similar to those for the discrete Fourier transform (DFT) are developed. The relations are valid if the filter frequency response is real and even. Two fairly simple relations are developed. The multiplication of the DCT of signal sequence and the DFT of filter sequence results in circular convolution of the folded signal sequence and the filter sequence. Thus, it can be used to filter an image in the frequency domain as an approximation of circular convolution in the spatial domain. >
73 citations
TL;DR: A novel Fourier technique for digital signal processing is developed based on the number-theoretic method of the Mobius inversion of series that competes with the classical FFT (fast Fourier transform) approach in terms of accuracy, complexity, and speed.
Abstract: A novel Fourier technique for digital signal processing is developed. This approach to Fourier analysis is based on the number-theoretic method of the Mobius inversion of series. The Fourier transform method developed is shown also to yield the convolution of two signals. A computer simulation shows that this method for finding Fourier coefficients is quite suitable for digital signal processing. It competes with the classical FFT (fast Fourier transform) approach in terms of accuracy, complexity, and speed. >
67 citations
TL;DR: It is shown that the complex phase of Q (sigma) is a key parameter which can be exploited to reduce significantly the thickness of the synthesized films and to control the shape of the refractive index profiles without affecting the spectral performance.
Abstract: Several errors inherent to the Fourier transform method for optical thin film synthesis, including the inaccuracy of the spectral functions Q (sigma) used in the Fourier transforms, are compensated numerically by using successive approximations. We show that the complex phase of Q (sigma) is a key parameter which can be exploited to reduce significantly the thickness of the synthesized films and to control the shape of the refractive index profiles without affecting the spectral performance. This method is compared to other well established thin film design techniques.
64 citations
TL;DR: In this article, the authors developed the circular convolution-multiplication relationship for the discrete cosine transform (DCT) similar to that of the discrete Fourier transform (DFT) for the filter frequency response.
63 citations
41 citations
TL;DR: The accuracy of two conjugate gradient fast Fourier transform formulations for computing the electromagnetic scattering by resistive plates of an arbitrary periphery is discussed and the former is found to be substantially more efficient.
Abstract: The accuracy of two conjugate gradient fast Fourier transform formulations for computing the electromagnetic scattering by resistive plates of an arbitrary periphery is discussed. One of the formulations is based on a discretization of the integral equations prior to the introduction of the Fourier transform, whereas the other is based on a similar discretization after the introduction of the Fourier transform. The efficiency and accuracy of these formulations are examined by comparison with measured data for rectangular and nonrectangular plates. The latter method is found to provide a more accurate computation of the plate scattering by eliminating aliasing errors (other than those due to undersampling). It is also found to be substantially more efficient. Its greatest advantage is realized when solving large systems generated by convolutional operators not yielding Toeplitz matrices, as is the case with plates having nonuniform resistivity. >
29 citations
TL;DR: In this paper, a new signal processing method was proposed for generating optimal stored wave form inverse Fourier transform (SWIFT) excitation signals used in FTMS or FT‐ICR.
Abstract: A new signal processing method has been proposed for generating optimal stored wave form inverse Fourier transform (SWIFT) excitation signals used in Fourier transform mass spectrometry (FTMS or FT‐ICR). The excitation wave forms with desired flat excitation power can be obtained by using the data processing steps which include: (1) smoothing of the specified magnitude spectrum, (2) generation of the optimal phase function, and (3) inverse Fourier transformation. In contrast to previously used procedures, no time domain wave form apodization is necessary. The optimal phase functions can be expressed as an integration of the specified power spectral profiles. This allows one not only to calculate optimal phase functions in discrete data format, but also to obtain an analytical expression (in simple magnitude spectral cases) that is for theoretical studies. A comparison is made of the frequency sweeping or ‘‘chirp’’ excitation and stored wave form inverse Fourier transform (SWIFT) excitation. This shows tha...
TL;DR: A technique is proposed for filtering multidimensional (MD) discrete signals that combines discrete Fourier transform (DFT) and linear difference equation (LDE) methods.
Abstract: A technique is proposed for filtering multidimensional (MD) discrete signals that combines discrete Fourier transform (DFT) and linear difference equation (LDE) methods. A partial P-dimensional DFT (P >
TL;DR: A computer-assisted method for analysis of multicomponent mixtures by use of conventional absorbance as well as discrete Fourier transforin coefficients (combined trigonometric functions) is presented.
TL;DR: In this paper, a new kind of Fourier transformation is proposed for distributions taking values in the Clifford algebra of three-dimensional space, with the unit imaginary replaced by the unit trivector.
Abstract: A new kind of Fourier transformation is proposed for distributions taking values in the Clifford algebra of three‐dimensional space, with the unit imaginary replaced by the unit trivector. The transformation is used to introduce special distributions that describe the free electromagnetic field.
TL;DR: In this paper, it was shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete Fourier transform (DFT) as long as such approximants exist.
Abstract: It is shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete Fourier transform (DFT) as long as such approximants exist. Based on this fact, it is permissible to apply the fast Fourier transform (FFT) algorithm in carrying out rational approximations without being apprehensive of convergence. The DFT is used to obtain traditional approximations for transfer functions of infinite-dimensional systems. Justification is provided for using the DFT in such approximations. It is established that whenever a stable transfer function can be approximated uniformly on the right half-plane by a rational function, its approximants can always be recognized by means of a DFT. >
Patent•
19 Apr 1990
TL;DR: In this paper, a discrete Fourier transform operation is performed by a CHIRP-Z transform or a Goertzel's second order Z-transform which can accommodate any number of data lines or values.
Abstract: Magnetic resonance imaging data lines or views are generated and stored in a magnetic resonance data memory (56). The number of views or phase encode gradient steps N along each of one or more phase encode gradient directions is selected (70) to match the dimensions of the region of interest. A discrete Fourier transform algorithm (94) operates on the data in the magnetic resonance data memory to generate an image representation for storage in an image memory (96). Unlike a fast Fourier transform algorithm which requires a N views or data lines, where a and N are integers, the discrete Fourier transform has a flexible number of data lines and data values which can be accommodated. More specifically to the preferred embodiment, the discrete Fourier transform operation is performed by a CHIRP-Z transform or a Goertzel's second order Z-transform which can accommodate any number of data lines or values.
TL;DR: A theorem concerning the least-squares projection of an arbitrary function onto an infinite basis of translated function is given, which provides an explicit formula for the Fourier transform, of the projected function.
Abstract: A theorem concerning the least-squares projection of an arbitrary function onto an infinite basis of translated function is given. The theorem provides an explicit formula for the Fourier transform, of the projected function. The formula has the advantage of being valid for least-squares projections in any N-dimensional space. The expression for the projected function can be approximately inverted, using the discrete Fourier transform, to find the actual basis coefficients. >
03 Apr 1990
TL;DR: In this paper, a system of retrieving the phase from a single interferogram with carrier frequency based on 2-D Fourier transform method is presented, where several procedures to modify an inter-ferogram and its spectrum in order to reduce the phase errors and detect the object domain are described.
Abstract: The system of retrieving the phase from a single interferogram with carrier frequency based on 2-D Fourier transform method is presented. Several procedures to modify an interferogram and its spectrum in order to reduce the phase errors and detect the object domain are described. The effects of refinements introduced are shown on experimental data.
TL;DR: Here it is shown that the CPFFT is the same as the SRFFT algorithm from the arithmetic complexity point of view.
Abstract: A recently introduced algorithm for the fast computation of the discrete Fourier transform ,called conjugate pair fast Fourier transform (CPFFT),seemed to require a smaller number of real multiplications and additions than that required for the split radix (SR) FFT algorithm.Here it is shown that the CPFFT is the same as the SRFFT algorithm from the arithmetic complexity point of view.
03 Apr 1990
TL;DR: It is shown that recognition accuracy can be maintained using only 26 b/frame as compared to the conventional calculation with floating-point accuracy, and the error rate can be even further reduced.
Abstract: The coefficients of the smoothed group delay spectrum (SGDS) are calculated by discrete-time Fourier transform of the linear prediction coefficients, i.e. the representation is in the frequency domain. Isolated word recognition experiments with a low bit quantization of these SGDS coefficients are reported. It is shown that recognition accuracy can be maintained using only 26 b/frame as compared to the conventional calculation with floating-point accuracy. Using a bark scale representation the error rate can be even further reduced. >
03 Apr 1990
TL;DR: An iterative algorithm that makes it possible to reconstruct the original signal from its Fourier intensity and the intensity of another signal that contains a known reference signal is presented.
Abstract: The problem of reconstructing either a one-dimensional or a two-dimensional signal from its Fourier intensity and the intensity of another signal that contains a known reference signal is addressed. After a brief summary of some conditions under which a one-dimensional or two-dimensional signal can be uniquely defined in terms of the given Fourier intensities, an iterative algorithm that makes it possible to reconstruct the original signal is presented. For the case of a point reference signal, a method is presented that can be used to determine the location and amplitude of the reference from the given Fourier intensity information. An example that illustrates the performance of the reconstruction algorithm is given. >
TL;DR: In this paper, the authors conclude the series with an examination of some of the most important types of two-dimensional spectra, and conclude with a discussion of their most important properties.
Abstract: This article will conclude the series with an examination of some of the most important types of two-dimensional spectra.
TL;DR: It is concluded that quantization “noise” is negligible; oversampling thus has little effect on FT spectral signal-to-noise ratio, dynamic range, or resolution; Oversampling can, however, improve phase and magnitude linearity by eliminating the need for a sharp cutoff in the passband of the analog filter.
Abstract: In Fourier transform spectrometry, an analog time-domain signal is sampled at equally spaced intervals and subjected to a discrete Fourier transform to yield a discrete frequency-domain spectrum. Round-off errors in the sampling process can generate quantization "noise" even for a noiseless time-domain analog signal. Oversampling refers to sampling a time-domain analog signal at a rate faster than that required by the Nyquist limit. Oversampling has been applied in a wide variety of fields, including image, speech, and audio spectral analysis. It has been variously claimed that oversampling can increase the effective number of analog-to-digital converter (ADC) bits, increase signal-to-noise ratio and/or resolution, allow for improved phase and/or magnitude linearity, and reduce quantization "noise" in the bandwidth of interest. In this paper, we explain and demonstrate the effects of oversampling in Fourier transform spectrometry. For Fourier transform interferometry, magnetic resonance, or ion cyclotron resonance mass spectrometry conducted with an ADC of at least 12 bit/word, we conclude that quantization "noise" is negligible; oversampling thus has little effect on FT spectral signal-to-noise ratio, dynamic range, or resolution. Oversampling can, however, improve phase and magnitude linearity by eliminating the need for a sharp cutoff in the passband of the analog filter. Finally, autocorrelation analysis of simulated time-domain signals shows that quantization "noise" is random and essentially independent of frequency (i.e., "white") at practically attainable sampling rates.
TL;DR: A FDRT-based algorithm is presented for computing 2-D DFTs, which has the advantages of having the lowest number of multiplications and being more suitable for parallel implementation compared with other related algorithms.
Abstract: The discrete Radon transform (DRT) has been known to convert two-dimensional discrete Fourier transforms (2-D DFTs) into 1-D DFTs. A fast discrete Radon transform (FDRT) algorithm is presented. A FDRT-based algorithm is presented for computing 2-D DFTs, which has the advantages of having the lowest number of multiplications and being more suitable for parallel implementation compared with other related algorithms.
03 Apr 1990
TL;DR: The performances of five discrete orthogonal transforms in speech encryption systems are compared and a figure of merit based on all the four objective measures is formed that gives good correlation to the subjective results of residual intelligibility and recovered speech quality.
Abstract: The performances of five discrete orthogonal transforms in speech encryption systems are compared. The transforms considered are the discrete Fourier transform, discrete cosine transform, Walsh-Hadamard transform, Karhunen-Loeve transform, and discrete prolate spheroidal transform. Four objective measures are used to grade the encryption systems with respect to residual intelligibility and recovered voice quality. A figure of merit based on all the four objective measures is formed. It gives good correlation to the subjective results of residual intelligibility and recovered speech quality. >
TL;DR: A procedure is developed for designing algorithms for one- and two-dimensional convolutions and discrete Fourier transforms using fast computational eigentransform procedures in different fields or rings (rational, real, and polynomial).
Abstract: A procedure is developed for designing algorithms for one- and two-dimensional convolutions and discrete Fourier transforms using fast computational eigentransform procedures in different fields or rings (rational, real, and polynomial). The high efficiency of the proposed algorithms is confirmed by numerical computations.
TL;DR: A fast algorithm is presented for numerical evaluation of forward and inverse Radon transforms by rewriting the transform as a convolution, a computational speed is obtained similar to the speed of the 2D fast Fourier transform.
Abstract: A fast algorithm is presented for numerical evaluation of forward and inverse Radon transforms. The algorithm does not perform exact one-to-one mapping as the discrete Fourier transform but, due to the use of band-limited basis functions, it is robust and sufficiently accurate for seismic applications. By rewriting the transform as a convolution, a computational speed is obtained similar to the speed of the 2D fast Fourier transform.
TL;DR: A simple 1-D and 2-D systolic array for realizing the discrete cosine transform (DCT) based on the discrete Fourier transform (DFT) fo an input sequence are presented.
Abstract: in this paper, simple 1-D and 2-D systolic array for realizing the discrete cosine transform (DCT) based on the discrete Fourier transform (DFT) fo an input sequence are presented. The proposed arrays are obtained by a simple modified DFT (MDFT) and an inverse DFT (IDFT) version of the Goertzel algorithm combined with Kung's approach. The 1-D array requiresN cells, one multiplier and takesN clock cycles to produce a completeN-point DCT. The 2-D array takes √N clock cycles, faster than the 1-D array, but the area complexity is larger. A continuous flow of input data is allowed and no idle time is required between the input sequences.
Patent•
07 Jun 1990TL;DR: In this article, the discrete Fourier transform is continuously calculated at input signal sample rate using recursive filtering, rather than transversal filtering, and the number of complex digital multiplications per computational cycle is reduced to 2N, where a triangular truncation window is used.
Abstract: The discrete Fourier transform is continuously calculated at input signal sample rate using recursive filtering, rather than transversal filtering. This reduces the number of complex digital multiplications per computational cycle to N, the number of spectral components in the discrete Fourier transform, where rectangular truncation window or a new exponential window is used. Where a triangular truncation window is used the number of complex digital multiplications per computational cycle is reduced to 2N.