Showing papers on "Linear approximation published in 1987"
TL;DR: The design of FIR digital filters with a complex-valued desired frequency response using the Chebyshev error is investigated, and the desired constant group delay is found to be smaller than that of a linear phase filter of the same length.
Abstract: The design of FIR digital filters with a complex-valued desired frequency response using the Chebyshev error is investigated. The complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem. A standard linear programming algorithm for the Chebyshev solution of overdetermined equations is used to solve the real approximation problem. Additional constraints are introduced which allow weighting of the phase and/or group delay of the approximation. Digital filters are designed which have nearly constant group delay in the passbands. The desired constant group delay which gives the minimum Chebyshev error is found to be smaller than that of a linear phase filter of the same length. These filters, in addition to having a smaller, approximately constant group delay, have better magnitude characteristics than exactly linear phase filters with the same length. The filters have nearly equiripple magnitude and group delay.
256 citations
TL;DR: In this article, the PartAN variant of the linear approximation method is adapted for solving the network equilibrium problem and its properties are analyzed by using algebraic and geometric approaches, and a simple and efficient algorithm is stated.
Abstract: The PARTAN variant of the linear approximation method is adapted for solving the network equilibrium problem. A simple and efficient algorithm is stated. Its properties are analyzed by using algebraic and geometric approaches. Its computational efficiency on small and large scale problems is compared to that of the linear approximation method.
84 citations
01 Dec 1987
TL;DR: The theoretical framework for finding such change of coordinates for a nonlinear system are described and a software package that symbolically solves these transformations is currently being prepared.
Abstract: One traditional approach in the analysis and design of nonlinear control systems is a first order approximation by a linear system. A new approach is to use nonlinear change of coordinates and feedback to construct linear approximations that are accurate to second and higher orders. However, the algebraic calculations required to obtain these aproximations are somewhat lengthy. In this paper, the theoretical framework for finding such change of coordinates for a nonlinear system are described. A software package that symbolically solves these transformations is currently being prepared.
80 citations
TL;DR: An algorithm for the problem of orthogonal ℓ1 fitting of discrete data is presented, along with numerical results of its application to some data sets.
Abstract: The problem is considered of orthogonal l1 fitting of discrete data. Local best approximations are characterized and the question of the robustness of these solutions is considered. An algorithm for the problem is presented, along with numerical results of its application to some data sets.
60 citations
TL;DR: In this article, two approaches for developing composition/depth profiles were examined and compared, and a method combining both models was presented as the best approach for efficiently generating nondestructive depth profiles.
55 citations
10 Jun 1987
TL;DR: In this article, the authors investigate the problem of accurately modeling nonlinear systems (such as aircraft flight in high angle-of-attack/sideslip flight) using simple low-order Volterra submodels.
Abstract: We investigate the problem of accurately modeling nonlinear systems (such as aircraft flight in high angle-of-attack/sideslip flight) using simple low-order Volterra submodels. First, we apply this technique to a simplified nonlinear stall/post-stall aircraft model for the case of a longitudinal limit cycle. Our simulation study demonstrates that the responses of the Volterra submodels accurately match the responses of the original nonlinear model, whereas the responses of a piecewise-linear model do not. Next, we apply the technique to a simplified high a nonlinear model of wing rock. Our simulation study demonstrates that the second-order Volterra approximation predicts the wing rock limit cycle, while a linear approximation does not. Third-, fourth- and fifth-order Volterra approximations are observed to give wing rock amplitudes that converge quadratically to the nonlinear value.
30 citations
01 Jan 1987
TL;DR: In this article, a most probable point-locus concept is proposed for probabilistic fracture mechanics analysis, where the initial locus is determined iteratively using the linear approximation of the service life function about the mean values of the random variables X.
Abstract: A systematic and efficient method for probabilistic fracture mechanics analysis is proposed. The method is based on a most-probable-point-locus concept. The locus is obtained iteratively where the initial locus is determined using the linear approximation of the service life N(X) function about the mean values of the random variables X. Linear and quadratic approximations of N(X) are established locally at the most probable points, and the reliability analysis methods are used to compute the cumulative probabilities. By using two examples, the proposed method is demonstrated to be efficient and accurate. One example involved a random loading and N(X) was computed using cycle-by-cycle integration. The method is general and can be applied to other performance functions. It is particularly suitable when the computation of the performance function is time consuming such that Monte Carlo simulation is prohibitively costly.
22 citations
TL;DR: In this article, an elegant and comprehensive treatment of image formation by a general optical system is shown to be possible by the use of Hamilton's method, and Lagrange's theorem is used to show how the eikonal, or angle characteristic function, can be calculated for a single refracting or reflecting surface.
Abstract: An elegant and comprehensive treatment of image formation by a general optical system is shown to be possible by the use of Hamilton’s method. Lagrange’s theorem is used to show how the eikonal, or angle characteristic function, can be calculated for a single refracting or reflecting surface and for an optical system composed of an arbitrary number of elements. We review the imaging properties of a general system in the linear approximation and show how the aberrations of second and third order can be calculated. We consider briefly the influence of symmetry on the type and number of aberrations. We give a simple deviation of the optical cosine condition of Smith and show how it is related to the imaging properties of the system.
18 citations
TL;DR: In this article, the equations of motion for the gravitational scattering of two point masses obtained in a post-linear approximation to general relativity are integrated by iteration starting with uniform straight-line motion.
Abstract: The equations of motion for the gravitational scattering of two point masses obtained in a post-linear approximation to general relativity are integrated by iteration starting with uniform straight-line motion. The ensuing conservation laws for energy and linear momentum disprove Rosenblum's recent claim of an energy loss in a post-linear approximation scheme.
17 citations
TL;DR: A nonlinear constrained optimizer, WATOPT, targeted towards circuit applications is described which utilizes a specially tailored quadratic program which reduces storage requirements and improves speed of computations.
Abstract: A nonlinear constrained optimizer, WATOPT, targeted towards circuit applications is described. It utilizes a specially tailored quadratic program which reduces storage requirements and improves speed of computations. Three advanced designs in the frequency domain show application.
16 citations
TL;DR: In this paper, it was shown that the nonlinear control process is globally null controllable with controlsu∈M(Ω) under the assumption that the linear approximation is stable.
Abstract: The question of controlling some nonlinear retarded functional differential equations from an initial function to the zero function is considered. The control setsM(Ω) are square integrable functions with values in the unit closed sphereL2([t0, ∞),Em) with center the origin. Assuming that the linear approximation of the nonlinear equation is null controllable with some integrable controls on some interval [t0,t1−2r], wheret0 is sufficiently large and wherer>0 is the delay, and assuming that the nonlinear system, withu=0, is uniformly globally asymptotically stable, we show that the nonlinear control process is globally null controllable with controlsu∈M(Ω). The paper gives conditions which guarantee the stability assumptions, and also indicates conditions which yield the null controllability assumptions of the linear approximation. Our research extends known results on ordinary differential processes.
01 Sep 1987
TL;DR: In the present research, an algorithm is developed which uses a polynomial approximation to f(A) by interpolating the function f(z) in a certain set of points which is known to have some maximal properties and is almost best.
Abstract: During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.
TL;DR: De Gruyter et al. as discussed by the authors used the linear formalism of non-equilibrium thermodynamics and found that apparent sedimentation coefficients (in the osmotically assisted sedimentation) are orders of magnitude larger than normal sedimentation coefficient.
Abstract: Solute sedimentation is accelerated if the solution or dispersion under study is allowed to settle in osmotic contact with the solvent, in a dialysis cell. By using the linear formalism of non-equilibrium thermodynamics it is found that apparent sedimentation coefficients (in the osmotically assisted sedimentation) are orders of magnitude larger than \"normal\" sedimentation coefficients. Moreover, the apparent sedimentation coefficients increase with increasing solute concentration, cell height and dialysis cell membrane area. These results may assist the experimenter's decisions regarding a choice between osmosedimentation and normal sedimentation techniques. Introduction Many chemical separation processes depend on mass flows arising from nonconjugate forces i.e., forces other than chemical potential gradients. In thermal osmosis [1 —4] and in thermal diffusion [5] the mass flow occurs as a response to temperature gradients. Mass and electrical charge flows also are strongly coupled in electrokinetic phenomena [1]· Two independent driving forces, the hydrostatic and the osmotic pressure, both contribute to the flow of the solvent in ultrafiltration and in reverse osmosis thus leading to coupled solvent flows. All of these phenomena are adequately described by Onsager's phenomenological relations [1]. The work performed in this laboratory in recent years showed that solute sedimentation rates are enhanced if a solution is left in contact with solvent through a vertical semipermeable membrane, under gravity or low inertial fields [6-9]. This effect allows a faster attainment of solution sedimentation equilibrium, even in long solution columns [8-10] and the determination of molecular-weights and virial coefficients of polymers in solution. Centrifugation of a solute within a dialysis cell (i.e., osmocentrifugation) does also allow the performance of useful J. Non-Equilib. Thermodyn,, Vol. 12, 1987, No. 3 Copyright © 1987 Walter de Gruyter · Berlin · New York 206 S. R Nunes, F. Galembeck preparative experiments, such as the concentration of macromolecular solutes [9] and the obtainment of density gradients (for cell separation) in low-speed centrifuges [11]. In this paper, osmocentrifugation is described by the linear approximation of non-equilibrium thermodynamics. Some relations are thus obtained which allow an effective comparison between centrifugation and osmocentrifugation experiments. 1. Theory The rate of approach to equilibrium, for any system, may be obtained by integrating the dissipation function over the system volume: dG ~~dt~ (1) For an isothermal system wherein no chemical reaction takes place, the dissipation function is = Ó / (2) where 7£ is the molar flow, i.e., the number of moles of component i crossing a unit area per unit time in a given direction and μί is the chemical potential of component i. The evaluation of Φ may be conveniently done by using the discontinuous system depicted in Fig. 1. It contains 4 compartments; compartments A and C (and Β and Z>) are connected by a semipermeable membrane and A and Β (and C and Z>) . V * ) ÷ ROTATION A X I S ,I
Patent•
02 May 1987
TL;DR: In this article, the authors proposed a method of least squares to combine broken line approximation, the integrated approximation, and the method-of- least squares (PMS) respectively.
Abstract: PURPOSE:To perform the approximation with high accuracy by combining successively the broken line approximation, the integrated approximation and the method of least squares respectively. CONSTITUTION:For the broken line approximation, a distance l is given previously to decide the degree of approximation and then compared with a distance dj from the segment connecting a start point A and an end point B. When the distance dj is larger than the distance l, a point C is connected to both end points by segments. Then the maximum distance is obtained between a straight line including a segment AC and every picture element held between both end points A and C of the segment. When said maximum distance is larger than the distance l, a segment is drawn in the same way to obtain the approximation. Hereafter the broken line approximation ACEB is obtained in the same way. For the integrated approximation, an approximate segment larger than the fixed length L is selected out of a series of broken lines. Then the start and end points are connected to each other between both segments to obtain an integrated segment as long as the distance between the extension of said approximate segment and the start or end point of the segment following said extension is smaller than a fixed value. Furthermore the method of least squares is applied to all points between the start and end points of each integrated segment for linear approximation. Then the straight line is cut at the positions closest to the start and end points respectively to obtain the segments to show pictures.
Patent•
22 Jul 1987
TL;DR: In this paper, a pipelined architecture linear approximation transform circuit using floating point number inputs, and slope (m) and intercept (b) information from lookup tables, for determining y = mx + b (where "b" can be negative or positive) to approximate a variety of curves by linear segments.
Abstract: A pipelined architecture linear approximation transform circuit using floating point number inputs, and slope (m) and intercept (b) information from lookup tables, for determining y = mx + b (where "b" can be negative or positive) to approximate a variety of curves by linear segments. Circuitry is also provided for rapidly calculating the square root or reciprocal of an input floating point number.
10 Jun 1987
TL;DR: In this paper, a new exact nonlinear filtering theory is applied to a practical radar tracking problem, where the standard extended Kalman filter for this application suffers from a fundamental flaw due to linearization.
Abstract: A new exact nonlinear filtering theory is applied to a practical radar tracking problem. The standard extended Kalman filter for this application suffers from a fundamental flaw due to linearization. The new nonlinear filter does not require this linear approximation.
01 Jan 1987
TL;DR: In this paper, a nonlinear system is approximated by a feedback linearizable system instead of a linear system and lower order invariants for the equivalence problem for two nonlinear systems are provided.
Abstract: We are interested in approximating a nonlinear system by a feedback linearizable system instead of a linear system Two approaches presently exist in the literature One involves the concept of involutivity to a certain order, and the other considers a canonical expansion and pure feedback approximation We show the relationship between these two methods This provides insight into lower order invariants in the equivalence problem for two nonlinear systems Moreover, the output time responses for a nonlinear system and its feed-back linearizable approximation are mentioned
TL;DR: The problem of maximizing the minimal mean square error is shown to be equivalent to solving a certain linear approximation problem whose solution can be characterized in such a way as to allow for an explicit calculation of it in special cases.
Abstract: In a communication system with sampling errors the worst case is considered where the minimal mean square error for a suitable choice of the transfer function becomes maximal. The transfer functions are allowed to be chosen from the space of all transfer functions of a given finite bandwidth which is a multiple of the Nyquist bandwidth. The problem of maximizing the minimal mean square error is shown to be equivalent to solving a certain linear approximation problem whose solution can be characterized in such a way as to allow for an explicit calculation of it in special cases.
TL;DR: In this article, the noise-sensitive hysteresis loops observed in the bouncing-ball model are described within the formalisms of the square map and the dissipative standard mapping.
Abstract: The noise-sensitive hysteresis loops observed in the bouncing-ball model are described. The phenomenon is analysed within the formalisms of the square map and the dissipative standard mapping. The notion of steady-state paths is introduced. A linear approximation of the simplest steady-state path is found.
TL;DR: In this paper, an equation analogous to the Khokhlov-Zabolotskaya equation was derived for narrow beams of quasitransverse waves propagating at small angles to a magnetic field.
Abstract: An equation, analogous to the Khokhlov-Zabolotskaya equation, is derived for narrow beams of quasitransverse waves propagating at small angles to a magnetic field. The effect of diffraction on wave propagation is investigated in the linear approximation.
TL;DR: In this article, the effects of perturbing norm and constraints in linear approximation were considered, and the effect of the perturbation norm on linear approximation was analyzed in the context of linear approximation.
TL;DR: A procedure is given for translating a rectangular grid in such a way that a finite set of points in ℝN is approximated as good as possible by points of the translated grid.
Abstract: A procedure is given for translating a rectangular grid in such a way that a finite set of points in ℝN is approximated as good as possible by points of the translated grid.
TL;DR: It is shown that the approximation can be a considerable improvement over the linear approximation and the asympototic stability of the closed-loop control system is established by Lyapunov’s approach.
01 Jan 1987
TL;DR: In this paper, an optical flow field arising from an irregular moving rigid surface was used to predict the performance of the algorithm in situations of practical interest, and the results showed that the approximation to epsilon ( nu ) is astonishingly accurate.
Abstract: The changes in an image arising from the motion of the environment have been intensively studied over the past few years. These changes, termed optical flow, depend on both the motion and the shape of the environment. The central problem in the study of optical flow is to find the best way of recovering information about the motion and shape of the environment from the associated optical flow field. The author has previously shown (1986, 1987) that some parameters of the motion can be recovered with greater accuracy than the remaining parameters, provided the optical flow field arises from an irregular moving rigid surface. A surface is classed as irregular if the set of inverse distances to points on the surface do not possess a good linear approximation. The algorithm in question is based on a least-square error function, epsilon ( nu ). In the case of an optical flow field arising from an irregular moving rigid surface, there in a simple approximation to epsilon ( nu ) which can be used to predict the performance of the algorithm in situations of practical interest. Experimental results show that the approximation to epsilon ( nu ) is astonishingly accurate. In this paper the approximation to epsilon ( nu ) is given and some experiments to check this approximation are described.
TL;DR: In this paper, the common properties of different classes of 2π-periodic solutions of Duffing's non-linear differential equation are investigated analytically and numerically, and the stability of these solutions in a linear approximation is also investigated.
Abstract: The common properties of different classes of 2π-periodic solutions of Duffing's non-linear differential equation are investigated analytically and numerically. Generating periodic solutions are examined. The basic properties of 2π-periodic solutions are investigated using the non-linear functional analysis method. Sets of 2π-periodic solutions of Duffing's equation are obtained using numerical methods for arbitrary values of the parameters occurring in it, and the stability of these solutions in a linear approximation is also investigated.
01 Oct 1987
TL;DR: In this article, the least squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal solutions to the problems of system compensation, inversion, and approximation.
Abstract: The least-squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal polynomial solutions to the problems of system compensation, inversion, and approximation. The polynomial coefficients are obtained from the solution to a so-called normal linear matrix equation, whose coefficients are shown to be the weighting patterns of certain linear systems. These, in turn, can be used in the recursive solution of the normal equation.
TL;DR: In this paper, the authors introduce the notion of Gaussian formulae and characterize the minimal representation for this approximation, which is based on linear combinations of integrals on manifolds of lower dimension contained in the region of integration.
TL;DR: In this paper, a method for determination of a linear approximation doping profile in a semiconductor is presented based on measurement of the high frequency C-V characteristic of a MOS capacitor.
Abstract: A method for determination of a linear approximation doping profile in a semiconductor is presented. The procedure is based on measurement of the high frequency C-V characteristic of a MOS capacitor. The proposed approximation of doping profile is sufficiently accurate in practical cases and the method is fast. Hence it can be applied in MOS device modelling or routine MOS technology diagnostics.