Showing papers on "Fractional Fourier transform published in 1990"
TL;DR: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied and the notion of time-frequency localization is made precise, within this framework, by two localization theorems.
Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >
6,180 citations
TL;DR: In this paper, the principal component transform (PCT) is applied to high-spectral-resolution remote-sensing data to transform the noise covariance matrix into the identity matrix, which is equivalent to the maximum noise fraction transform.
Abstract: High-spectral-resolution remote-sensing data are first transformed so that the noise covariance matrix becomes the identity matrix. Then the principal components transform is applied. This transform is equivalent to the maximum noise fraction transform and is optimal in the sense that it maximizes the signal-to-noise ratio (SNR) in each successive transform component, just as the principal component transform maximizes the data variance in successive components. Application of this transform requires knowledge or an estimate of the noise covariance matrix of the data. The effectiveness of this transform for noise removal is demonstrated in both the spatial and spectral domains. Results that demonstrate the enhancement of geological mapping and detection of alteration mineralogy in data from the Pilbara region of Western Australia, including mapping of the occurrence of pyrophyllite over an extended area, are presented. >
468 citations
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
102 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
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: In this paper, a procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent.
Abstract: Walsh functions are a set of complete and orthonormal functions of nonsinusoidal waveform. In contrast to sinusoidal waveforms whose amplitudes may assume any value between -1 to +1, Walsh functions assume only discrete amplitudes of + or -1 which form the kernel function of the Walsh transform. Because of this special nature of the kernel, computation of the Walsh transform of a given signal is simpler and faster than that of the Fourier transform. The properties of the Fourier transform in linear time are similar to those of the Walsh transform in dyadic time. The Fourier transform has been widely used in interpretation of geophysical problems. Considering various aspects of the Walsh transform, an attempt has been made to apply it to some gravity data.A procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent. The technique has been applied to data from the published literature to evaluate its applicability, and the results are in good agreement with the more conventional ones.
59 citations
03 Apr 1990
TL;DR: The discretization is developed, leading to a form which can run with any FFT (fast Fourier transform) routine and the advantage of the technique is illustrated by computing broadband radar ambiguity functions and affine time-frequency representations.
Abstract: Theoretical wideband studies generally provide expressions involving stretched forms of the signal. This feature complicates implementation of the results and suggests the use of a Mellin transform to process dilations efficiently. A tool for the practical development of this idea is given. The definition, properties, and time-frequency interpretation of the relevant Mellin transform are given. The discretization is developed, leading to a form which can run with any FFT (fast Fourier transform) routine. The advantage of the technique is illustrated by computing broadband radar ambiguity functions and affine time-frequency representations. >
43 citations
41 citations
01 Nov 1990
TL;DR: In this article, a new technique for fringe-pattern analysis is described, based on the γHilbert transform. But the technique is computationally inefficient. And it requires a large number of inputs.
Abstract: A new technique for fringe-pattern analysis is described. The technique, based on the
Hilbert transform, is highly accurate and computationally efficient.
TL;DR: The exponential Radon transform on the plane arises in single photon emission computed tomography as discussed by the authors, which differs from the usual Radon transformation by an exponential weight along the line of integration.
Abstract: The exponential Radon transform on the plane arises in single photon emission computed tomography. It differs from the usual Radon transform by an exponential weight along the line of integration. The full description of the image of the transform in the space of compactly supported smooth functions is given. This description is connected with some curious identities for the sin function.
TL;DR: The fast algorithm for the (real) Hartly transform is discussed in relation to the established fast algorithmFor the (complex) Fourier transform, compared by timing comparably written programs on a given machine, and the discipline of timing is discussed as an adjunct to complexity analysis.
Abstract: The fast algorithm for the (real) Hartly transform is discussed in relation to the established fast algorithm for the (complex) Fourier transform. The two transforms are compared by timing comparably written programs on a given machine, and the discipline of timing is discussed as an adjunct to complexity analysis. With real data, one Hartley transform program can economically replace such packages as a complex-valued unilateral Fourier transform combined with a real-valued unilateral inverse Fourier transform. The Hartley transform is favorable for fast convolution of real data sets. The utility of spectral analysis into Fourier series throughout physics suggested that the Hartley transform might have less physical significance, but the construction of Hartley diffraction planes in the microwave and optical laboratories, where electromagnetic phase is encoded as real-valued field amplitudes, has revealed interesting complementarity. >
01 May 1990
TL;DR: Efficient recursive methods and circuits for computing a continuously updated discrete Fourier transform of an input-digital signal are considered, based on the interpretation of the FT as a set of simultaneous bandpass filters.
Abstract: Efficient recursive methods and circuits for computing a continuously updated discrete Fourier transform (DFT) of an input-digital signal are considered. The Fourier transform (FT) is recomputed at each sample input time, with only O(N) operations being required to compute the transform, where N is the number of frequency bins. Various window functions are considered for windowing the input-wave form, namely a rectangular window, a triangular window, and an exponential window. The last type of window has not been widely considered in the past, partly due to its asymmetrical shape, and hence nonlinear phase response. Nevertheless, it is shown to have certain advantages in ease of computation and in flexibility. For the exponential window, a circuit that conveniently allows zooming in to particularly interesting parts of the frequency spectrum is shown. By appropriately loading a multiplier storage RAM, arbitrarily fine resolution may be achieved in any part of the spectrum, thus permitting closely adjacent peaks to be distinguished. The general approach is based on the interpretation of the FT as a set of simultaneous bandpass filters. Though these filters are generally finite-impulse-response filters, computational advantages are derived from formulating them as recursive filters. >
TL;DR: For Riemannian symmetric spaces (RSS) of non-compact type Helgason [2, this article found the analog of classical Fourier analysis for RSS of compact type.
Abstract: For Riemannian symmetric spaces (RSS) of noncompact type Helgason [2], [3], [5], found the analog of classical Fourier analysis. This paper concerns the counterpart of Helgason's theory for RSS of compact type. Together with classical Fourier theory these results constitute a unified Fourier analysis on RSS related to, but distinct from the established alternatives of representation theory and the spherical Fourier transform. An advantage of this style of Fourier theory is that the transform kernel has the same kind of simplicity as the functions e/xy of classical Fourier theory. In particular, the transform kernel employs scalar-valued eigenfunctions of first order differential operators. On the other hand, the theory given here for the compact RSS involves a severe singularity in part of the transform kernel. One may avoid this singularity by confining consideration to the half of the RSS closest to the origin; call this the local theory. The local theory is given in Section 1 for any compact RSS. The rest of this paper is devoted to the global theory for compact RSS of rank one. (It is not clear that a global theory exists for the higher rank compact RSS.) In broad outline, Helgason's transform comes from the wedding of the spherical Fourier transform with an integral formula for the Poisson kernel. In Helgason's notation ([5], p. 418) this formula is
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. >
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: The Hartley transform (HT) as discussed by the authors is an integral transform similar to the Fourier transform (FT), and it has most of the characteristics of the FT. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT).
Abstract: The Hartley transform (HT) is an integral transform similar to the Fourier transform (FT). It has most of the characteristics of the FT. Several authors have shown that fast algorithms can be constructed for the fast Hartley transform (FHT) using the same structures as for the fast Fourier transform. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT). Consequently, any process requiring the DFT (such as amplitude and phase spectra) can be performed faster by using the DHT. The general properties of the DHT are reviewed first, and then an attempt is made to use the FHT in some seismic data processing techniques such as one‐dimensional filtering, forward seismic modeling, and migration. The experiments show that the Hartley transform is two times faster than the Fourier transform.
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 >
03 Apr 1990
TL;DR: The fast approximated discrete transform is proposed as a method for reducing the time necessary to compute the discrete transform of a finite-length sequence by discarding the computations in bands that have little or no energy contribution.
Abstract: The fast approximated discrete transform is proposed as a method for reducing the time necessary to compute the discrete transform of a finite-length sequence. It is based on a subband decomposition and can be viewed as a link between the fast transform methods (like the fast Fourier transform), which compute all points in the transform domain, and the variety of methods to evaluate the discrete transforms at a given set of points. The method uses knowledge about the input signal to obtain an approximation to its transform by discarding the computations in bands that have little or no energy contribution. In a number of practical cases the proposed fast approximation is reasonably accurate, and in all cases the method can be iterated to yield the exact transform, if necessary. >
TL;DR: An interpretation of the Frei-Chen masks in terms of eight-dimensional Fourier transform coefficient vectors is introduced and a modified set of eight orthogonal masks based on the frequency space analysis is proposed.
01 May 1990
TL;DR: In this paper, a simple method for the design of a digital all-pass filter, satisfying the given group delay specification, is presented, based on the discrete Hilbert transform relation, relating the log-magnitude and phase of the Fourier transform of the minimum phase signal.
Abstract: A simple method for the design of a digital all-pass filter, satisfying the given group delay specification, is presented. The design is based on the discrete Hilbert transform relation, relating the log-magnitude and phase of the Fourier transform of the minimum phase signal. The transfer function of an all-pass filter is completely determined by the coefficients of the denominator polynomial. For the filter to be stable, the denominator polynomial must be minimum phase. From the given group delay specification, the phase corresponding to the pole part of the desired filter is first determined. The magnitude spectrum corresponding to the pole part of the desired filter is obtained from the above phase through the discrete Hilbert transform relation. The method needs just four fast Fourier transform operations. There is no restriction on the order of the filter, and the number of filter coefficients can be selected after the final design, depending on the accuracy desired. The procedure is illustrated through design examples. >
01 Aug 1990
TL;DR: In this article, an uncertainty inequality for the Fourier transform on the Heisenberg group was shown to be equivalent to the classical uncertainty inequalities for the Euclidean Fourier transformation.
Abstract: We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.
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: The linear complexityL2(G) of a finite groupG is the minimal number of additions, subtractions and multiplications by complex constants of absolute value ≦2 sufficient to evaluate a suitable Fourier transform of ℂG.
Abstract: The linear complexityL2(G) of a finite groupG is the minimal number of additions, subtractions and multiplications by complex constants of absolute value ≦2 sufficient to evaluate a suitable Fourier transform of ℂG. Combining and modifying several classical FFT-algorithms, we show thatL2(G)≦8|G|log2|G| for any finite metabelian groupG.
01 Nov 1990
TL;DR: In this article, the relationship between the coefficients of the Fourier expansion of a periodic signal and the wavelet expansion of the same signal was derived, and the relationship was used to approximately calculate the coefficients.
Abstract: This paper derives the relationship between the coefficients of the Fourier expansion of a periodic signal and the coefficients of the wavelet expansion of the same signal. The formula derived shows how the Fourier concept of frequency and the wavelet concept of scale are related and how the wavelet coefficients display the information contained in the signal in a new way. The relationship also shows how the wavelet expansion can be used to approximately calculate the Fourier coefficients.
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. >
21 Mar 1990
TL;DR: This study concentrates on discrete orthogonal transforms such as the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT) for low-rate seismic data compression.
Abstract: An investigation of low-rate seismic data compression using transform techniques is presented. This study concentrates on discrete orthogonal transforms such as the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT). Uniform and subband transform coding schemes are implemented, and comparative results are given for data rates ranging from 150 to 550 b/s. Results are also compared with existing linear prediction techniques. >
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.