Showing papers on "Fractional Fourier transform published in 1991"
TL;DR: In this article, a constant Q transform with a constant ratio of center frequency to resolution has been proposed to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components.
Abstract: The frequencies that have been chosen to make up the scale of Western music are geometrically spaced. Thus the discrete Fourier transform (DFT), although extremely efficient in the fast Fourier transform implementation, yields components which do not map efficiently to musical frequencies. This is because the frequency components calculated with the DFT are separated by a constant frequency difference and with a constant resolution. A calculation similar to a discrete Fourier transform but with a constant ratio of center frequency to resolution has been made; this is a constant Q transform and is equivalent to a 1/24‐oct filter bank. Thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range. This transform against log (frequency) to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted. This is compared to the conventio...
890 citations
TL;DR: The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing D FTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies.
Abstract: This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{ - 2\pi i\alpha } $ where $\alpha $ is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.
344 citations
Journal Article•
TL;DR: In this article, a canonical decomposition for abelian schemes over a smooth quasiprojective base 5 over a field has been established for Chow ring tensored with Q. In order to formulate such a statement, Manin's paper [Ma] was extended.
Abstract: such that the /-adic realization of h (A) is H (A, Qt). Recall that Chow motives are obtained from the category of smooth projective varieties over a field by a construction of Grothendieck using äs intersection theory the Chow ring tensored with Q. See [Ma] and [Mur], 1.1 and l .6, remark 2 for details. By an intricate argument Shermenev also shows how to express h *(A) in terms of h (A). For Chow motives a decomposition äs in (0.1) with /-adic realization äs above is by no means unique. Shermenev's decomposition in particular depends on choices. In this paper we establish a canonicalfunctorial decomposition äs above not only for abelian varieties but also for abelian schemes over a smooth quasiprojective base 5 over a field. In order to formulate such a Statement we extend parts of Manin's paper [Ma] on motives over a field to the case where the base is S. This is straightforward and we obtain a category of relative Chow motives over S.
263 citations
TL;DR: In this paper, the authors present an English translation of the Beurling Theorem, which states that a function and its Fourier transform cannot simultaneously be very small at infinity.
Abstract: There are many theorems known which state that a function and its Fourier transform cannot simultaneously be very small at infinity, such as various forms of the uncertainty principle and the basic results on quasianalytic functions. One such theorem is stated on page 372 in volume II of the collected works of Arne Beurling [1]. Although it is not in every respect the most precise result of its kind, it has a simplicity and generality which make it very attractive. The editors state that no proof has been preserved. However, in my files I have notes which I made when Arne Beurling explained this result to me during a private conversation some time during the years 1964---1968 when we were colleagues at the Institute for Advanced Study, I shall reproduce these notes here in English translation with onIy minor details added where my notes are too sketchy. Theorem. Let fELl (R) and assume that
163 citations
TL;DR: In this article, the submanifolds under consideration were shown to have a (L, LP) -restriction theorem that holds for 1 < p < 3 f, where f is the number of vertices in an n-dimensional Euclidean space.
Abstract: In this note we will prove a (L , LP) -restriction theorem for certain submanifolds & of codimension / > 1 in an n— dimensional Euclidean space which arise as orbits under the action of a compact group K. As is well known such a result can in general only hold for 1 < p < j^y. We will show that for the submanifolds under consideration the inequality ^\\f(x)\\dKx)
130 citations
TL;DR: In this article, Chandra, Helgason, Trombi and Varadarajan give a short proof of their results, based on the Paley-Wiener theorem.
85 citations
TL;DR: A linear systolic array for the discrete cosine transform, discrete sine transform and their inverses is developed, which has the advantages of pipelinability, regularity, locality, and scalability, making it quite suitable for VLSI signal processing.
Abstract: A linear systolic array for the discrete cosine transform, discrete sine transform, and their inverses is developed. It generates the transform kernel values recursively. Compared to the scheme with the transform kernel values prestored in memory either inside or outside each processing element, the clock period is shortened by a memory access time. In addition, the array pays no cost for prestorage. The systolic array has the advantages of pipelinability, regularity, locality, and scalability, making it quite suitable for VLSI signal processing. >
77 citations
Proceedings Article•
01 Jan 1991TL;DR: In this paper, a parallel algorithm for the Fourier transform on the star graph is presented, which requires O(n/sup 2/) multiply-add steps for an input sequence of n! elements, and is hence cost-optimal with respect to the sequential algorithm on which it is based.
Abstract: The n-star graph, denoted by S/sub n/, is one of the graph networks that have been recently proposed as attractive alternatives to the n-cube topology for interconnecting processors in parallel computers. We present a parallel algorithm for the computation of the Fourier transform on the star graph. The algorithm requires O(n/sup 2/) multiply-add steps for an input sequence of n! elements, and is hence cost-optimal with respect to the sequential algorithm on which it is based. This is believed to be the first algorithm, and the only one to date, for the computation of the Fourier transform on the star graph. >
74 citations
TL;DR: It is shown that a DCT coding scheme reproduces faithfully the seismic waveform at approximately one-third of the original rate, revealing that sinusoidal transforms are well-suited for robust, low-rate seismic signal representation.
Abstract: The authors consider the development and evaluation of transform coding algorithms for the storage of seismic signals. Transform coding algorithms are developed using the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT). These are evaluated and compared to a linear predictive coding algorithm for data rates ranging from 150 to 550 bit/s. The results reveal that sinusoidal transforms are well-suited for robust, low-rate seismic signal representation. In particular, it is shown that a DCT coding scheme reproduces faithfully the seismic waveform at approximately one-third of the original rate. >
74 citations
TL;DR: A vector transform, originally developed for digital filtering, is used and it is shown that this vector transform does have the decorrelation and energy-packing properties.
Abstract: A vector transform is introduced. The application of the vector transform to image coding is discussed. The development of a vector transform coding technique consists of two parts. One part is to find a vector transform that has the decorrelation and energy-packing properties. The other part is to find a coding algorithm in the vector transform domain. A vector transform, originally developed for digital filtering, is used. It is shown that this vector transform does have the decorrelation and energy-packing properties. Codebook design algorithms for the transform domain vectors are discussed. An implementation scheme for the vector transform is presented. >
69 citations
TL;DR: In this paper, a new family of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles is developed, and the first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.
Abstract: Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.
Book•
01 Jan 1991TL;DR: In this paper, the authors present a survey of the use of the Fourier Transform for continuous-time and Discrete-Time Signals and its application in a variety of systems.
Abstract: (NOTE: Each chapter begins with an Introduction and ends with a Summary and Problems). 1. Overview of Signals and Systems. Signals. Systems. 2. Continuous-Time and Discrete-Time Signals. PART A Continuous-Time Signals. Basic Continuous -Time Signals. Modification of the Variable t. Continuous-Time Convolution. PART B Discrete-Time Signals. Basic Discrete-Time Signals. Modification of the Variable n. Discrete-Time Convolution. 3. Linear Time-Invariant Systems. PART A Continuous-Time Systems. System Attributes. Continuous-Time LTI Systems. Properties of LTI Systems. Differential Equations and Their Implementation. PART B Discrete-Time Systems. System Attributes. Discrete-Time LTI Systems. Properties of LTI Systems. Difference Equations and the Their Implementation. 4. Fourier Analysis for Continuous-Time Signals. The Eigenfunctions of Continuous-Time LTI Systems. Periodic Signals and the Fourier Series. The Continuous-Time Fourier Transform. Properties and Applications of the Fourier Transform. APPLICATION 4.1 Amplitude Modulation. APPLICATION 4.2 Sampling. 5.Frequency Response of LTI Systems. APPLICATION 4.3 Filtering. 5. The Laplace Transform. The Region of Convergence. The Inverse Laplace Transform. Properties of the Laplace Transform. The System Function for LTI Systems. Differential Equations. APPLICATION 5.1 Butterworth Filters. Structures for Continuous-Time Filters. Appendix 5A The Unilateral Laplace Transform. Appendix 5B Partial-Fraction Expansion for Multiple Poles. 6. The z Transform. The Eigenfunctions of Discrete-Time LTI Systems. The Region of Convergence. The Inverse z Transform. Properties of the z Transform. The System Function to LTI Systems. Difference Equations. APPLICATION 6.1 Second-Order IIR Filters. APPLICATION 6.2 Linear-Phase FIR Filters. Structures for Discrete-Time Filters. APPENDIX 6A The Unilateral z Transform. APPENDIX 6B Partial-Fraction Expansion for Multiple Poles. 7. Fourier Analysis for Discrete-Time Signals. The Discrete-Time Fourier Transform. 2.Properties of the DTFT. APPLICATION 7.1 Windowing. 3.Sampling. 4.Filter Design by Transformation. 5.The Discrete Fourier Transform/Series. APPLICATION 7.2 FFT Algorithm. 8. State Variables. Discrete-Time Systems. Continuous-Time Systems. Operational-Amplifier Networks. Bibliography. Index.
TL;DR: In this paper, the effect of weighting on the uncertainty of the discrete time Fourier transform (DTFT) samples of a signal corrupted by additive noise is investigated, and it is shown how the adopted window sequence and the autocovariance function of the noise affect the second-order stochastic moments of the frequency domain data.
Abstract: The effect of weighting on the uncertainty of the discrete time Fourier transform (DTFT) samples of a signal corrupted by additive noise is investigated. Making very weak assumptions, it is shown how the adopted window sequence and the autocovariance function of the noise affect the second-order stochastic moments of the frequency-domain data. The relationship obtained extends the results reported in the literature and is useful in many frequency-domain estimation problems. It is shown how the knowledge of the second-order moments of the transform has allowed the application of the least squares technique for the estimation of the parameters of a multifrequency signal in the frequency-domain. The estimator obtained is very useful when high-accuracy results are required under real-time constraints. The procedure exhibits a better accuracy than similar frequency-domain methods proposed in the literature. >
TL;DR: In this article, the authors compare scalare and vector transform methods for global spectral models of the shallow-water equations and show that the vector transform method is algebraically equivalent to the more usual scalar transform methods.
Abstract: We compare scalare and vector transform methods for global spectral models of the shallow-water equations. For the scalar transform methods, we demonstrate some economies in the number of Legendre transforms required. It is shown that the vector transform method is algebraically equivalent to the more usual scalar transform methods, and the choice of transform grid is discussed.
TL;DR: An efficient algorithm is developed for computing the continuous wavelet transform or wideband ambiguity function on a grid whose samples are spaced uniformly in time but placed arbitrarily in scale.
Abstract: An efficient algorithm is developed for computing the continuous wavelet transform or wideband ambiguity function on a grid whose samples are spaced uniformly in time but placed arbitrarily in scale. The method is based on the chirp z transform and requires the same order of computation as constant-bandwidth analysis techniques, such as the shorttime Fourier transform and narrowband ambiguity function.
01 Jan 1991
TL;DR: These notes are an introduction to basic concepts and tools in group representation theory that are fundamental for the analysis of radar and sonar imaging that should be easily comprehensible by engineers and physicists, as well as mathematicians.
Abstract: These notes are an introduction to basic concepts and tools in group representation theory, both commutative and noncommutative, that are fundamental for the analysis of radar and sonar imaging. Several symmetry groups of physical interest are treated (circle, line, rotation, ¢ ¡ ¤ £ ¦ ¥ , Heisenberg, etc.) together with their associated transforms and representation theories (DFT, Fourier transform, expansions in spherical harmonics, wavelets, etc.). Through the unifying concepts of group representation theory, familiar tools for commutative groups, such as the Fourier transform on the line, extend to transforms for the noncommutative groups which arise in radar-sonar. The insight and results obtained will are related directly to objects of interest in radar-sonar, such as the ambiguity function. The material is presented with many examples and should be easily comprehensible by engineers and physicists, as well as mathematicians.
14 Apr 1991
TL;DR: In this paper, an extended lapped transform (ELT) based on a type-IV discrete cosine transform (DCT-IV) is proposed for AR (autoregressive) signals and also for real speech signals.
Abstract: The author presents a fast algorithm for extended lapped transform (ELT), which is a modulated lapped transform (MLT) with longer basis functions. The proposed algorithm is based on a type-IV discrete cosine transform (DCT-IV). For AR (autoregressive) signals and also for real speech signals, the coding performance of the ELT is shown to be significantly higher than that of a block transform such as the DCT (discrete cosine transform), actually approaches the performance of ideal filter banks. Therefore, the ELT is a promising substitute for traditional block transforms in transform coding systems, and also a good substitute for less efficient filter banks in subband coding systems. >
TL;DR: It is shown that it is possible to replace the real-numbered elements of a discrete cosine transform (DCT) matrix with integers and still maintain the structure, i.e., relative magnitudes and orthogonality, among the matrix elements.
Abstract: It is shown that it is possible to replace the real-numbered elements of a discrete cosine transform (DCT) matrix with integers and still maintain the structure, i.e., relative magnitudes and orthogonality, among the matrix elements. The result is an integer cosine transform (ICT). Thirteen ICTs have been found, and some of them have performance comparable to the DCT. The main advantage of the ICT lies in having only integer values, which in two cases can be represented perfectly by 6-bit numbers, thus providing a potential reduction in the computational complexity. >
TL;DR: In this paper, the authors derived formulas for the two-dimensional Fourier transform of functions with polygonal support and linear amplitude variation from the corresponding formula for a constant function, valid for all nonzero values of the transform variable k, which fail when k is perpendicular or parallel to any edge of the polygon.
Abstract: New formulas for the two-dimensional Fourier transform of functions with polygonal support and linear amplitude variation are derived from the corresponding formula for a constant function. These expressions, valid for all nonzero values of the transform variable k, are superior to those previously reported, which fail when k is perpendicular or parallel to any edge of the polygon. These transforms have applications in diffraction theory and computational electromagnetics. >
Patent•
14 Nov 1991
TL;DR: In this article, a signal processing apparatus and method for iteratively determining the inverse Arithmetic Fourier Transform (AFT) of an input signal by converting the input signal, which represents Fourier coefficients of a function that varies in relation to time, space, or other independent variable, into a set of output signals representing the values of a Fourier series associated with the input signals.
Abstract: A signal processing apparatus and method for iteratively determining the inverse Arithmetic Fourier Transform (AFT) of an input signal by converting the input signal, which represents Fourier coefficients of a function that varies in relation to time, space, or other independent variable, into a set of output signals representing the values of a Fourier series associated with the input signal. The signal processing apparatus and method utilize a process in which a data set of samples is used to iteratively compute a set of frequency samples, wherein each computational iteration utilizes error information which is calculated between the initial data and data synthesized using the AFT. The iterative computations converge and provide AFT values at the Farey-fraction arguments which are consistent with values given by a zero-padded Discrete Fourier Transform (DFT), thus obtaining dense frequency domain samples without interpolation or zero-padding.
TL;DR: In this paper, the direct scattering transform (DST) is used to analyze nonlinear wave data in the finite-line KdV domain. But the authors assume that the wave amplitude is recorded in the form of a discrete space or time series which is determined either by experimental measurement or by computer simulation of the physical system of interest.
TL;DR: It is shown that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG and it is proved that these Fourier transforms are efficient to evaluate.
Abstract: The linear complexityL
K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L
K(G):= min{L
K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L
K(G)≤8.5|G|log|G| for any supersolvable groupG andL
K(G)≤1.5|G|log|G| for any 2-groupG.
TL;DR: In this paper, the relationship between NMR spectra obtained by the maximum entropy method and by conventional processing (Fourier transformation) is explored, and it is argued that a maximum entropy reconstruction can reveal information that could not be obtained from a single Fourier transform spectrum.
TL;DR: The accuracy of the output of the Fast Fourier Transform is studied by estimating the expectedvalue and the variance of the accompanying linear forms in terms of the expected value and variance ofThe relative roundoff errors for the elementary operations of addition and multiplication.
Abstract: We study the accuracy of the output of the Fast Fourier Transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication. We compare the results with the corresponding ones for the direct algorithm for the Discrete Fourier Transform, and we give indications of the relative performances when different rounding schemes are used. We also present the results of numerical experiments run to test the theoretical bounds and discuss their significance.
Patent•
22 Nov 1991TL;DR: In this paper, a method for performing a discrete cosine transform on a transform input value is described, where a plurality of predetermined transform coefficients are provided by truncating and modifying prior art transform coefficients.
Abstract: A method is disclosed for performing a discrete cosine transform on a transform input value wherein the discrete cosine transform has a plurality of predetermined transform coefficients. A number N 1 of shift operations is determined independently of the transform input value in order to provide a set of N 1 of shift operations. A number N 2 of add operations is determined independently of the transform input value in order to provide a set of N 2 add operations. The transform input value is operated upon only by the N 1 shift operations and the N 2 add operations to provide a discrete cosine transform output value without any multiplication. This method may be applied to both forward and inverse discrete cosine transforms. The transform coefficients are simplified coefficients which are provided by truncating and modifying prior art transform coefficients. This simplification is adapted to provide coefficients which require fewer than a predetermined number of shift and add operations in order to determine an approximation of the product which would result from a multiplication by the coefficient. The simplification of the coefficients causes degradation of a video image transformed using the simplified coefficients. Therefore the simplification is constrained to cause an acceptable amount of image degradation.
TL;DR: In this article, a 3D Fourier transform of a linear function with triangular support is derived in its coordinate-free representation, which is applied to the near-field computation, yielding accurate numerical solutions.
Abstract: A three-dimensional Fourier transform (FT) of a linear function with triangular support is derived in its coordinate-free representation. The Fourier transform of this distribution is derived in three steps. First, the 2-D FT of a constant (top hat) function is obtained. Next, the distribution is generalized to a linearly varying function. Finally, the formulation is extended to a coordinate-free representation which is the 3-D FT of the 2-D function defined over a surface. This formulation is applied to the near-field computation, yielding accurate numerical solutions. >
Patent•
NEC1
TL;DR: In this paper, a linear transform calculation on a product signal produced by multiplying a predetermined transform window function and an apparatus input signal is described. But this is not applicable to either of forward and inverse transform units.
Abstract: In an apparatus for carrying out a linear transform calculation on a product signal produced by multiplying a predetermined transform window function and an apparatus input signal, an FFT part (23) carries out fast Fourier transform on a processed signal produced by processing the product signal in a first processing part (21). As a result, the FFT part produces an internal signal which is representative of a result of the fast Fourier transform. A second processing part (22) processes the internal signal into a transformed signal which represents a result of the linear transform calculation. The apparatus is applicable to either of forward and inverse transform units (11, 12).
TL;DR: In this paper, an analysis of the discrete Fourier transform of a monotonically decaying function, represented by a sum of exponentials with negative exponents, is presented, which is particularly applicable to stress relaxation and creep in viscoelastic materials.
Abstract: An analysis is presented of the discrete Fourier transform of a monotonically decaying function, represented by a sum of exponentials with negative exponents. The results are compared with those of a previous analysis based on the analytical Fourier transform, which proposed a method for extending the frequency range of the Fourier transform of experimental data by combining transforms performed over different time periods. The principle of the method is confirmed but comparison shows that results derived for the analytical transform cannot always be applied directly to the discrete transform. Modifications are therefore proposed which improve the accuracy and mitigate the aliasing effects evident in short-time transforms while keeping the computing time to a minimum. These results are particularly applicable to stress relaxation and creep in viscoelastic materials and an example from articular cartilage shows the compliance modulus over a frequency range from 10-3 Hz to 230 Hz from one creep experiment of duration 18.7 min.
TL;DR: A new procedure that uses only a single image is developed that estimates the locations of real-plane zeros from either the measured Fourier modulus data or a stagnated reconstruction and uses this information in the iterative Fourier transform algorithm to force the complex-valued Fourier data to have real- plane zeros at the correct locations.
Abstract: The iterative Fourier transform algorithm, although it has been demonstrated to be a practical phase retrieval algorithm, suffers from certain stagnation problems. Specifically, there exists a stripe stagnation problem, in which stagnated reconstructed images exhibit stripelike features throughout the image, which is particularly difficult to overcome. Previous solutions to this problem used multiple reconstructions and did not address the cause. In this paper a new procedure that uses only a single image is developed that estimates the locations of real-plane zeros from either the measured Fourier modulus data or a stagnated reconstruction and uses this information in the iterative Fourier transform algorithm to force the complex-valued Fourier data to have real-plane zeros at the correct locations. It is shown that this procedure overcomes the stripe stagnation.
TL;DR: In this paper, the Hartley transform is used to calculate transients and nonsinusoidal waveshape propagation in electric power systems, and a general introduction to the use of Hartley transforms for electric circuit analysis is presented.
Abstract: Because the Fourier transform causes the convolution operation to become a simple complex product, it has been used to solve power system problems. A similar convolution property of the Hartley transform is used to calculate transients and nonsinusoidal waveshape propagation in electric power systems. The importance of this type of calculation relates to the impact of loads, particularly electronic loads, whose demand currents are nonsinusoidal. An example is given in which the Hartley transform is used to assess the impact of an electronic load with a demand which contains rapidly changing current. The authors also present a general introduction to the use of Hartley transforms for electric circuit analysis. A brief discussion of the error characteristics of discrete Fourier and Hartley solutions is presented. Because the Hartley transform is a real transformation, it is more computationally efficient then the Fourier or Laplace transforms. >