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Showing papers on "Fourier series published in 1985"


Book
01 Aug 1985
TL;DR: These twenty lectures have been developed and refined by Professor Siebert during the more than two decades he has been teaching introductory Signals and Systems courses at MIT and are designed to familiarize students with the properties of a fundamental set of analytical tools.
Abstract: These twenty lectures have been developed and refined by Professor Siebert during the more than two decades he has been teaching introductory Signals and Systems courses at MIT. The lectures are designed to pursue a variety of goals in parallel: to familiarize students with the properties of a fundamental set of analytical tools; to show how these tools can be applied to help understand many important concepts and devices in modern communication and control engineering practice; to explore some of the mathematical issues behind the powers and limitations of these tools; and to begin the development of the vocabulary and grammar, common images and metaphors, of a general language of signal and system theory.Although broadly organized as a series of lectures, many more topics and examples (as well as a large set of unusual problems and laboratory exercises) are included in the book than would be presented orally. Extensive use is made throughout of knowledge acquired in early courses in elementary electrical and electronic circuits and differential equations.Contents: Review of the "classical" formulation and solution of dynamic equations for simple electrical circuits; The unilateral Laplace transform and its applications; System functions; Poles and zeros; Interconnected systems and feedback; The dynamics of feedback systems; Discrete-time signals and linear difference equations; The unilateral Z-transform and its applications; The unit-sample response and discrete-time convolution; Convolutional representations of continuous-time systems; Impulses and the superposition integral; Frequency-domain methods for general LTI systems; Fourier series; Fourier transforms and Fourier's theorem; Sampling in time and frequency; Filters, real and ideal; Duration, rise-time and bandwidth relationships: The uncertainty principle; Bandpass operations and analog communication systems; Fourier transforms in discrete-time systems; Random Signals; Modern communication systems."Circuits, Signals, and Systems" is included in The MIT Press Series in Electrical Engineering and Computer Science, copublished with McGraw-Hill.

351 citations



Journal ArticleDOI
TL;DR: In this article, a multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations, and a selection criterion is given for determining the optimal number of harmonics to be included.
Abstract: Results involving correlation properties and parameter estimation for autoregressive-moving average models with periodic parameters are presented. A multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations. Close approximation to the likelihood function for Gaussian PARMA processes results in efficient maximum-likelihood estimation procedures. Terms in the Fourier expansion of the parameters are sequentially included, and a selection criterion is given for determining the optimal number of harmonics to be included. Application of the techniques is demonstrated through analysis of a monthly streamflow time series.

190 citations


Journal ArticleDOI
TL;DR: Perpendicular distance line transect models are examined to assess whether any single model can provide a general procedure for analysing line Transect data, and the hazard-rate model appears promising, whereas the exponential power series and exponential quadratic models do not.
Abstract: SUMMARY Perpendicular distance line transect models are examined to assess whether any single model can provide a general procedure for analysing line transect data. Of the two-parameter models considered, the hazard-rate model appears promising, whereas the exponential power series and exponential quadratic models do not. Of the nonparametric models, the Fourier series is the best developed, and is favoured by many researchers as a general model. However, for a given data set, the Fourier series estimate may be highly dependent on the number of terms selected, and so the model is not a clear improvement over the hazard-rate model. A similar variable-term model, using Hermite polynomials, is considered, and is shown to be less dependent on the number of terms selected. There has been some debate about whether the derivative of the density function of perpendicular distances evaluated at 0 should be 0, so that the function has a "shoulder." The problem is examined in detail, and it is argued that reliable estimation is not possible from line transect data unless a shoulder exists. Many data sets appear to exhibit no shoulder; possible reasons are examined.

137 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the composition of the Fourier transform in Rn followed by restriction to r defines a bounded operator from LP(Rn) to Lq(F) for certain p, q. The curvature hypotheses are the weakest under which this could hold, and p is optimal for a range of q.
Abstract: For smooth curves F in Rn with certain curvature properties it is shown that the composition of the Fourier transform in Rn followed by restriction to r defines a bounded operator from LP(Rn) to Lq(F) for certain p, q. The curvature hypotheses are the weakest under which this could hold, and p is optimal for a range of q. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.

136 citations


Journal ArticleDOI
TL;DR: In this paper, the mixed-boundary value problem for a typical strip is reduced to a singular integral equation, which is solved numerically by the use of Fourier series techniques, and a check of the accuracy of the computations is provided by the balance of rates of energies.

99 citations


Journal ArticleDOI
TL;DR: In this article, a general expression of the Fourier operational matrix of integration P is derived which is analogous to that previously derived for other types of orthogonal functions such as Walsh, block-pulse, Laguerre, Legendre and Chebyshev.
Abstract: A general expression of the Fourier operational matrix of integration P is derived which is analogous to that previously derived for other types of orthogonal functions such as Walsh, block-pulse, Laguerre, Legendre and Chebyshev. This matrix P may be used to solve problems like identification, analysis and optimal control.

92 citations


Journal ArticleDOI
TL;DR: In this article, a Fourier series window approach is employed in conjunction with surface coils to accomplish one-dimensional spatial localization of phosphorus-containing metabolites in phantoms and the in vivo rat brain.

83 citations


Journal ArticleDOI
TL;DR: Aanonsen et al. as discussed by the authors considered nonlinear propagation of a periodic sound beam in a dissipative fluid using Fourier series expansion and numerical methods to solve the governing equation of motion in the parabolic approximation.
Abstract: Nonlinear propagation of a periodic sound beam in a dissipative fluid is considered using Fourier series expansion and numerical methods to solve the governing equation of motion in the parabolic approximation. The nearfield was considered in a previous paper [Aanonsen et al., J. Acoust. Soc. Am. 75, 749–768 (1984)]. The analysis is now extended to the farfield. Numerical and asymptotic results are derived and used to explain the development of the fundamental and harmonic components from the nearfield into the farfield. A discussion is also given of some earlier models for the farfield of directional waves. Emphasis is put on the importance of imposing the proper matching conditions between the nearfield solution and the spherical solution in the farfield in order to obtain a good approximation. Propagation and saturation curves are calculated, as well as beam patterns for various harmonic components. The results are compared with available experimental observations. Nonlinear effects, although generated...

80 citations


Journal ArticleDOI
TL;DR: In this article, the selection of an optimal parametric angle θ describing a closed magnetic flux surface is considered with regard to accelerating the convergence rate of the Fourier series for the Cartesian coordinates x(θ,φ)≡R−R0 and y(φ, φ) ≥Z−Z0.
Abstract: The selection of an optimal parametric angle θ describing a closed magnetic flux surface is considered with regard to accelerating the convergence rate of the Fourier series for the Cartesian coordinates x(θ,φ)≡R−R0 and y(θ,φ)≡Z−Z0. A system of algebraic equations, which are quadratic in the Fourier amplitudes of x and y, is derived by minimizing the width of the surface power spectrum. The solution of these nonlinear equations, together with the prescribed surface geometry, determines a unique optimal angle. A variational principle is used to solve these constraint equations numerically. Application to the representation of three‐dimensional magnetic flux surfaces is considered.

79 citations


Journal ArticleDOI
01 Apr 1985
TL;DR: In this paper, the local approximation order from a scale (S sub h) of approximating functions on R to the m power is characterized in terms of the linear span (and its Fourier transform) of the finitely many compactly supported functions.
Abstract: : Document characteristics the local approximation order from a scale (S sub h) of approximating functions on R to the m power is characterized in terms of the linear span (and its Fourier transform) of the finitely many compactly supported functions psi (Author)

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed and further developed the adiabatic invariance method for computing semiclassical eigenvalues, which was recently introduced by Solov'ev.
Abstract: In this paper we analyze and further develop the adiabatic invariance method for computing semiclassical eigenvalues. This method, which was recently introduced by Solov’ev, is basically an application of the Ehrenfest adiabatic hypothesis. The eigenvalues are determined from a classical calculation of the energy as the time dependent Hamiltonian H(t)=H0+s(t)H1 is switched adiabatically from the separable reference Hamiltonian H0 to the system Hamiltonian H0+H1. A systematic study is carried out to determine the best form for the switching function, s(t), to maximize the rate of convergence of the energy to its adiabatic limit. Five switching functions, including the linear function used by Solov’ev, are defined and tested on three different systems. The linear function is found to have a very slow convergence rate compared to the others. The classical energy is shown to be a periodic function of the angle coordinates of H0. The coefficients of the Fourier series representation of this function are then s...

Journal ArticleDOI
TL;DR: In this article, the conditions générales d'utilisation (http://www.numdam.org/conditions) are defined, i.e., toute utilisation commerciale ou impression systématique is constitutive d'une infraction pénale.
Abstract: © Annales de l’institut Fourier, 1985, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Journal ArticleDOI
TL;DR: In this article, a problem-oriented method for the nonlinear elastic-plastic analysis of nonsymmetrically loaded shells of revolution is described, based on an approximation of the circumferential distribution of the loads and variables by Fourier series.

Journal ArticleDOI
TL;DR: In this paper, a new method of estimating the truncation point which minimizes the mean integrated squared error (MISF) of a Fourier series density estimator is proposed.
Abstract: A new method of estimating the truncation point which minimizes the mean integrated squared error (MISF) of a Fourier series density estimator is proposed The similarity of the new method to Akaike’s (1974) AIC procedure for estimating the dimension of a statistical model suggests a second method of choosing a truncation point.It is shown that both of the new methods lead to consistent estimation of the underlying density.A simulation study indicates that for certain density functions the new methods are an improvement over the original method of Kronmal and Tarter (1968).

Journal ArticleDOI
TL;DR: In this article, strong approximation of Fourier series is shown to be the best possible refinement of an inequality of L. Leindler, which is related to the inverse problem.


Journal ArticleDOI
TL;DR: The convolution is defined as the sum of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian as discussed by the authors.
Abstract: The convolution is defined as the sum Open image in new window where Open image in new window for n≠0 Open image in new window and W0,W1 are arbitrary smooth functions Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function

Journal ArticleDOI
TL;DR: In this article, the authors introduce the generalized Bohr theorem and the singularity of homeomorphisms of the circle, and the correction problem and the space of these singularities.
Abstract: CONTENTSIntroduction § 1. The correction problem and the space § 2. Irremovable singularities § 3. Homeomorphisms of the circle: Bohr's theorem and Luzin's problem § 4. The generalized Bohr theorem § 5. Singularity of homeomorphisms transforming into § 6. SupplementsReferences

Journal ArticleDOI
TL;DR: In this article, a Bayesian approach to choosing between two non-nested multivariate regression systems is developed, which involves the calculation of the posterior probabilities of alternative hypotheses and formation of a posterior odds ratio.

Journal ArticleDOI
TL;DR: In this article, the principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire.

Journal ArticleDOI
TL;DR: A technique for automatic inspection of machine part cross-sections by using a planar closed curve representing the boundary profile of a manufactured machine part to obtain dimensional error between the measured and the theoretical parts.
Abstract: A technique for automatic inspection of machine part cross-sections is presented. A planar closed curve representing the boundary profile of a manufactured machine part is compared with the theoretical part. The boundary profile of the machine part is sampled by a high precision coordinate measurement machine which is capable of contour tracing. A curve fitting scheme using complex trigonometric functions (Fourier series) is used to model the machine part boundary contour. An upper bound for the accuracy of approximation of the arc length parametrized contours is suggested and compared with the upper bound given in ( Comput. Graphics Image Process. 6 , 1977, 277–285). Normalized Fourier coefficients are used to obtain dimensional error between the measured and the theoretical parts. Through an interactive procedure, critical locations are inspected for dimensional accuracy. Using a least squares minimizing technique, partially measured contours are matched with the theoretical contours.

Journal ArticleDOI
TL;DR: In this article, the authors treated the problem of shape identification of ferromagnetic materials as a shape identification problem with the use of Fourier descriptors and proposed a shape-based representation of the B/H plane behavior.
Abstract: In this paper the task of modeling the B/H plane behavior of ferromagnetic materials is treated as a shape identification problem solvable with the use of Fourier descriptors. This representation is found to be parsimonious in terms of data storage requirements and allows direct calculation of hysteresis losses.

Journal ArticleDOI
TL;DR: In this article, a fully discretized projection method with Fourier series is proposed, which is based on a modification of the fast Fourier transform and is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves.
Abstract: Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.

Journal ArticleDOI
01 Feb 1985
TL;DR: On considere la convergence de la serie de Fourier generalisee ΣΠ(x(g))u(g) dans le produit croise d'une algebre de von Neumann par un groupe discret, on demontre que cette serie converge dans une topologie introducedite par Bures as mentioned in this paper.
Abstract: On considere la convergence de la serie de Fourier generalisee ΣΠ(x(g))u(g) dans le produit croise d'une algebre de von Neumann par un groupe discret. On demontre que cette serie converge dans une topologie introduite par Bures


Journal ArticleDOI
TL;DR: In this article, the analytical and numerical properties of the Fourier transform of a two-center product of exponentially declining functions (exponential-type functions, ETFs) are derived with the help of Fourier convolution theorem and Feynman's identity.


Journal ArticleDOI
TL;DR: In this article, the inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered.
Abstract: The inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered. In the limiting process we find that the widths of the forbidden bands approach the amplitudes of the Fourier spectrum. The number of spectral bands is estimated from Fourier theory and depends explicitly on the assumed spatial discretization in the wave amplitude function (potential). This allows one to estimate the number of degrees of freedom in a discrete (and, therefore, finite-banded) potential. An essential feature of the calculations is that all results for the periodic problem are cast in terms of the infinite-line reflection and transmission coefficientsb(k), a(k). Thus the connection between the whole-line and periodic problems is clear at every stage of the computations.

Journal ArticleDOI
TL;DR: In this paper, an experimental device for the dynamic determination of Mueller matrices which employs two rotating compensators and two fixed analyzers is described, which can be suitably arranged for the study of transmitting, reflecting or scattering samples.
Abstract: An experimental device for the dynamic determination of Mueller matrices which employs two rotating compensators and two fixed analyzers is described. The rotating compensators are equivalent quarter wave plates designed by the authors. The system generates a periodic intensity signal, which is then Fourier analyzed. The Fourier coefficients contain information about the Mueller matrix elements. The experimental device can be suitably arranged for the study of transmitting, reflecting or scattering samples.