DOCSLIB.ORG

Numerical Generation of Images for the Gibbs Phenomenon Near a Corner in the Plane Alia Khurram, Member, IAENG, and David W Kammler

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numerical Generation of Images for the Gibbs Phenomenon Near a Corner in the Plane Alia Khurram, Member, IAENG, and David W Kammler IAENG International Journal of Applied Mathematics, 44:1, IJAM_44_1_03 ______________________________________________________________________________________ Numerical Generation of Images for the Gibbs Phenomenon Near a Corner in the Plane Alia Khurram, Member, IAENG, and David W Kammler Abstract—In this paper we have studied the Gibbs ripples corner in the case where the function is doubly periodic. associated with an α-corner, i.e., the indicator function for the Helmberg’s approximation deletes all components that lie R2 smaller region of bounded by two rays from the origin that outside of a square in frequency space. In contarst, we intersect at the angle α, 0 < α < π. (Such corners cannot occur in the univariate case.) We have done this by removing from remove high frequency components by deleting from the the Fourier representation all components that lie outside of a Fourier representation all terms having an index that exceeds disk with radius σ ≥ 0. We have used a technique to rearrange some threshold. the approximating integrals in a form that facilitates efficient computation. We then produced two dimensional images for II. THE GIBBS PHENOMENON FOR THE HEAVISIDE STEP the Gibbs ripples that correspond to the corner angles α = R1 π=3, π=2, 2π=3, π. We observed that the maximum overshoot IN occurs near the corner. A suitably regular function f(x) on R has the Fourier representation Index Terms—Gibbs Phenomena, Fourier Transform, Bivari- Z ate Analysis. 1 f(x) = F (s)e2πisx ds ; (1) s=−∞ I. INTRODUCTION where the Fourier transform F is given by Z T the close of the nineteenth century the physicists A.A. 1 −2πisx Michelson and S.W. Stratton constructed a mechanical F (s) := f(x)e dx : (2) A −∞ harmonic analyzer that could plot graphs for trigonometric x= j j polynomials with up to 80 terms [11]. After producing If we suppress all frequencies s < σ, we obtain the low pass approximation many plots of partial sums for Fourier series of periodic Z σ functions with jump discontinuities, they observed that all 2πisx such approximations exhibit annoying “ripples” near every fσ(x) := F (s)e ds ; σ > 0 : (3) s=−σ point of discontinuity. J.W. Gibbs provided an informal We can use the unit box analysis of this phenomenon [4] which now bears his name ( [5], [6]. An earlier exposition was given by Wilbraham [7] 1 if −1=2 < s < 1=2 in 1841. rect(s) := (4) 0 otherwise, Gibbs ripples are produced by removing the high fre- quency components from the Fourier representation of a to remove the high frequency components from (1). Indeed, since ( function that has a jump discontinuity. If the function is ( ) s 1 if jsj < σ periodic, this involves truncating the corresponding Fourier rect = (5) series. If the function is aperiodic, this involves a similar 2σ 0 otherwise, truncation of the Fourier integral representation. This phe- we see that nomenon has been studied in several dimensions [2], [9]. An Z 1 ( ) evident manifestation of the phenomenon is a rippling effect s 2πisx fσ(x) = F (s) rect e ds : (6) which is undesirable in some applications. For example, in s=−∞ 2σ reconstruction of magnetic resonance imaging (MRI) data, The functions F (s) and rect(s=2σ) are the Fourier transforms methods are being developed to remove the Gibbs ringing of f(x) and 2σ sinc(2σx), respectively, so we can use the artifact that occurs at the boundaries of tissues [1], [8]. convolution rule for Fourier transforms to see that Moving from unvariate to bivariate functions, there are several important new features to consider. If the function has fσ(x) = f(x) ∗ 2σ sinc(2σx) ; a jump discontinuity across a smooth curve in its domain, one or equivalently might reasonably expect to produce Gibbs ripples analogous Z 1 − to those from the univariate case. A less predictable phe- sin[2πσ(x u)] fσ(x) = f(u) − du: (7) nomenon occurs when the function has a jump discontinuity u=−∞ π(x u) across a corner where two smooth curves in the domain We are interested in studying the behavior of such a low intersect at some angle α, 0 < α < π. A recent paper of pass approximation near a point of jump discontinuity, so we G. Helmberg [6] analyzes the Gibbs phenomenon at such a will replace f by the Heaviside step function 8 > Manuscript received July 24, 2013; revised November 30, 2013. <>1 if x > 0 A. Khurram is in the Mathematics Department at Adrian College, Adrian, MI, 49221 USA e-mail: ([email protected]). h(x) = >1=2 if x = 0 (8) David Kammler is professor emiritus at Southern Illinois University :> Carbondale, Carbondale, IL 62901. 0 if x < 0: (Advance online publication: 13 February 2014) IAENG International Journal of Applied Mathematics, 44:1, IJAM_44_1_03 ______________________________________________________________________________________ 1.2 where the bivariate Fourier transform is given by Z Z 1 1 1 −2πi(ux+vy) 0.8 F (u; v) := f(x; y)e dydx: x=−∞ y=−∞ 0.6 (14) 2 0.4 We will suppress all frequencies u; v outside the disk u + 2 2 0.2 v ≤ σ to obtain the low pass approximation ZZ 0 f (x; y) := F (u; v) e2πi(ux+vy) dvdu: (15) −0.2 σ −10 −5 0 5 10 u2+v2≤σ2 x We will use the unit diameter cylinder Fig. 1. The low pass approximation g(x) = hσ( 2σ ) for the Heaviside step (8) ( 1 if u2 + v2 < 1=4 cyl(u; v) := (16) 0 otherwise, The low pass approximation (7) for this function is Z 1 to write sin[2πσ(x − u)] ZZ ( ) hσ(x) = du: (9) u v 2πi(ux+vy) π(x − u) fσ(x; y) = F (u; v) cyl ; e dvdu : u=0 2σ 2σ We shall find it convenient to work with the dilate uv Z (17) 1 − x sin[π(x 2σu)] The radially symmetric function cyl(x; y) is the hσ( ) = − 2σdu 2σ u=0 π(x 2σu) Fourierp transform of the radially symmetric function Z 1 sin[π(x − u)] jinc( x2 + y2), where = du ; π(x − u) ( u=0 1 if r = 0 which is independent of σ. With this in mind, we define the jinc(r) := univariate Gibbs function J1(π r)=2r if r > 0, Z 1 x sin[π(x − u)] and where, J1 is the first order Bessel function [3], p.359. g(x) := h ( ) = du: (10) Z σ 2σ π(x − u) 1 u=0 −2πix ir sin 2πx J1(r) := e e dx; The oscillating integrand in (10) decreases like 1=u 0 as u ! 1. Thus the improper integral converges, but it u v Since the functions F (u; v) and cyl( 2σ ;p2σ ) are the Fourier converges much too slowly to evaluate by direct numerical transforms of f(x; y) and (2σ)2 jinc (2σ x2 + y2), respec- integration. We take x > 0 and use the known definite tively, we can use the two dimensional convolution rule to integral [10] p. 62 Z Z see that 0 1 p sin(πu) sin(πu) 1 2 2 2 du = du = ; (11) fσ(x; y) = f(x; y) ∗ (2σ) jinc (2σ x + y ); u=−∞ πu u=0 πu 2 or equivalently to rewrite (10) in the form ZZ p Z Z − 2 − 2 x − 1 − J1[2πσ (x p) + (y q) ] sin[π(x u)] sin[π(x u)] fσ(x; y) = σf(p; q) p dqdp: g(x) = − du + − du (x − p)2 + (y − q)2 u=0 π(x u) u=x π(x u) pq Z Z 1 x sin(πu) sin(πu) (18) = du + du We are interested in studying the behavior of such a low u=0Z πu u=0 πu 1 x pass approximation near “edges”and “corners” where f has = + sinc(u) du: 2 a jump discontinuity. With this in mind, we select a corner u=0 angle α with 0 < α < 2π and define the α-corner function (12) 8 We can use numerical quadrature to evaluate the remaining > >1 if −α=2 < arg(x; y) < α=2 integral in (12). > > ± <>1=2 if arg(x; y) = α=2 The Gibbs function g is shown in Fig. (1). The character- 6 Cα(x; y) := > and (x; y) = (0; 0) (19) istic Gibbs ripples are evident. The function g has maxima > > of 1:089; 1:033; 1:020; ::: at x = 1; 3; 5; ::: respectively and >α=2π if (x; y) = (0; 0) :> minima of 0:951, 0:974, 0:983; ::: at x = 2; 4; 6; :::, respec- 0 otherwise. tively. The maximum Gibbs overshoot is a bit less than 9% of the unit jump in h. Here arg(x; y) is the smaller angle between the ray joining (0; 0) to (x; y) and the positive x-axis as shown in Fig. 2. 2 2 C (x; y) R III. THE GIBBS PHENOMENON FOR AN α-CORNER IN R Thus α is the indicator function of a wedge from that has the corner angle α (with suitable average values on We will now extend the analysis from the previous section the edge and vertex, cp.(8).) to a two dimensional context. A suitably regular bivariate It can be seen from (19) and Fig. 2 that two such corner f R2 function on has the Fourier representation functions with vertex angles α and 2π − α can be combined Z 1 Z 1 2πi(ux+vy) as f(x; y) = F (u; v)e dvdu (13) − u=−∞ v=−∞ Cα(x; y) + C2π−α( x; y) = 1: (20) (Advance online publication: 13 February 2014) IAENG International Journal of Applied Mathematics, 44:1, IJAM_44_1_03 ______________________________________________________________________________________ IV.
Load more

Recommended publications
  • GIBBS PHENOMENON SUPPRESSION and OPTIMAL WINDOWING for ATTENUATION and Q MEASUREMENTS* Cheh Pan Department of Geophysics, Stanfo
    SLAC-PUB-6222 September 1993 (Ml GIBBS PHENOMENON SUPPRESSION AND OPTIMAL WINDOWING FOR ATTENUATION AND Q MEASUREMENTS* Cheh Pan Department of Geophysics, Stanford University Stanford, CA 94305-2215 and Stanford Linear Accelerator Center Stanford, CA 94309 INTRODUCTION There are basically four known techniques, spectral ratios (Hauge 1981, Johnston and Toksoz 1981, Moos 1984, Goldberg et al 1985, Patten 1988, Sams and Goldberg 1990), forward modeling (Chuen and Toksoz 1981), inversion (Cheng et al 1986), and first pulse rise time (Gladwin and Stacey 1974, Moos 1984), that have been used in the past to measure the attenuation coefficient or quality factor from seismic and acoustic data. Problems have been encountered in using the spectral techniques, which included: (1) the correction for geometrical divergence of the acoustic wave front; (2) the suppression of the , Gibbs phenomenon or the ringing effect in the spectra; and (3) the elimination of contamination from interfering wave modes. Geometrical corrections have been presented by Patten (1988) and Sams and Goldberg (1990) for the borehole acoustics case. The remaining difficulties in the application of the spectral ratios technique come mainly from the suppression of the Gibbs phenomenon and optimal windowing of wave modes. This report deals with these two problems. e - Submitted to Geophysics * Work supP;orted by Department of Energy contract DE-ACO3-76SFOO5 15 : I FUNDAMENTALS The amplitudes R&j) of a seismic signal at frequencyfrecorded by a receiver at a distance or offset x from the source can be represented as, R(x,f)= A(f)G( (1) where A(/) is the source term, G(x) is the geometrical divergence which is assumed to be independent of frequency, and a is the attenuation coefficient.
    [Show full text]
  • Homework 1 Due February 2
    Math 262 / CME 372 Winter 2021 Homework 1 Due February 2 1. Gibbs Phenomenon. Gibbs' phenomenon has to do with how poorly Fourier series converge in the vicinity of a jump or discontinuity of a signal f. This fact was pointed out by Gibbs in a letter to Nature (1899). (Actually Gibb's phenomenon was first described by the British mathematician Wilbraham (1848).) The function Gibbs considered was a sawtooth (1). Gibbs was replying to a letter by the physicist Michaelson to Nature (1898), in which the latter expressed himself doubtful as to the idea that a real discontinuity (in f) can a replace a sum of continuous curves (Sn(f)). Recall that the Fourier series of a periodic function f on the unit interval is Z 1 X i2πkt −i2πkt f(t) = ck e with ck = f(t)e dt: 0 k2Z To investigate Gibb's phenomenon, let us look at the function on the unit interval ( t 0 ≤ t < 1=2 f(t) = (1) t − 1 1=2 ≤ t < 1 (a) Calculate the Fourier coefficients (ck) of f. P i2πkt (b) Let Sn(f) be the partial sum Sn = jk|≤n cke . Calculate the approximation error kf − Sn(f)kL2 as accurately as possible. (c) Consider now −1 0 ≤ t < 1=2 f(t) = : 1 1=2 ≤ t < 1 Gibbs observed that in the vicinity of the jump, the partial sums always overshoot the mark by about 9%. Verify this assertion by carefully setting up a numerical experiment. (d) Repeat the last question on the sawtooth signal (1).
    [Show full text]
  • Fourier Analysis
    FOURIER ANALYSIS Lucas Illing 2008 Contents 1 Fourier Series 2 1.1 General Introduction . 2 1.2 Discontinuous Functions . 5 1.3 Complex Fourier Series . 7 2 Fourier Transform 8 2.1 Definition . 8 2.2 The issue of convention . 11 2.3 Convolution Theorem . 12 2.4 Spectral Leakage . 13 3 Discrete Time 17 3.1 Discrete Time Fourier Transform . 17 3.2 Discrete Fourier Transform (and FFT) . 19 4 Executive Summary 20 1 1. Fourier Series 1 Fourier Series 1.1 General Introduction Consider a function f(τ) that is periodic with period T . f(τ + T ) = f(τ) (1) We may always rescale τ to make the function 2π periodic. To do so, define 2π a new independent variable t = T τ, so that f(t + 2π) = f(t) (2) So let us consider the set of all sufficiently nice functions f(t) of a real variable t that are periodic, with period 2π. Since the function is periodic we only need to consider its behavior on one interval of length 2π, e.g. on the interval (−π; π). The idea is to decompose any such function f(t) into an infinite sum, or series, of simpler functions. Following Joseph Fourier (1768-1830) consider the infinite sum of sine and cosine functions 1 a0 X f(t) = + [a cos(nt) + b sin(nt)] (3) 2 n n n=1 where the constant coefficients an and bn are called the Fourier coefficients of f. The first question one would like to answer is how to find those coefficients.
    [Show full text]
  • ON the GIBBS PHENOMENON and ITS RESOLUTION 1. Introduction. the Gibbs Phenomenon, As We View It, Deals with the Issue of Recover
    SIAM REV. c 1997 Society for Industrial and Applied Mathematics Vol. 39, No. 4, pp. 644–668, December 1997 004 ON THE GIBBS PHENOMENON AND ITS RESOLUTION∗ DAVID GOTTLIEB† AND CHI-WANG SHU† Abstract. The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local infor- mation from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting. The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation. Key words. Gibbs phenomenon, Galerkin, collocation, Fourier, Chebyshev, Legendre, Gegen- bauer, exponential accuracy AMS subject classifications. 42A15, 41A05, 41A25 PII. S0036144596301390 1. Introduction. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. A prototype problem can be formulated as follows: ˆ Given 2N +1 Fourier coefficients fk, for N k N, of an unknown function f(x) defined everywhere in 1 x 1, construct,− ≤ accurately,≤ point values of the function.
    [Show full text]
  • Higher-Order Wavelet Reconstruction/Differentiation Filters and Gibbs Phenomena
    Journal of Computational Physics 305 (2016) 244–262 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp Higher-order wavelet reconstruction/differentiation filters and Gibbs phenomena Richard Lombardini 1, Ramiro Acevedo 2, Alexander Kuczala 3, Kerry P. Keys 4, ∗ Carl P. Goodrich 5, Bruce R. Johnson Department of Chemistry, Smalley-Curl Institute and Laboratory for NanoPhotonics, Rice University, Houston, TX 77005, USA a r t i c l e i n f o a b s t r a c t Article history: An orthogonal wavelet basis is characterized by its approximation order, which relates to Received 5 December 2014 the ability of the basis to represent general smooth functions on a given scale. It is known, Received in revised form 18 October 2015 though perhaps not widely known, that there are ways of exceeding the approximation Accepted 23 October 2015 order, i.e., achieving higher-order error in the discretized wavelet transform and its inverse. Available online 30 October 2015 The focus here is on the development of a practical formulation to accomplish this Keywords: first for 1D smooth functions, then for 1D functions with discontinuities and then for Wavelets multidimensional (here 2D) functions with discontinuities. It is shown how to transcend Projection both the wavelet approximation order and the 2D Gibbs phenomenon in representing Reconstruction electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the Gibbs wavelet-basis grid. Multidimensional © 2015 Elsevier Inc. All rights reserved. Boundary 1. Introduction Wavelets are of general interest in developing systematically-improvable multiscale methods for solving differential equa- tions in quantum mechanics, electromagnetism and many other applications.
    [Show full text]
  • The Gibbs-Phenomenon in Option Pricing Methods
    Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics The Gibbs phenomenon in option pricing methods Filtering and other techniques applied to the COS method A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by Mark Versteegh Delft, Netherlands October 2012 Copyright c 2012 by Mark Versteegh. All rights reserved MSc thesis APPLIED MATHEMATICS The Gibbs phenomenon in option pricing methods Filtering and other techniques applied to the COS method MARK VERSTEEGH Delft University of Technology Daily Supervisors Responsible professor Prof.dr.ir. C.W. Oosterlee Prof.dr.ir. C.W. Oosterlee M.J. Ruijter, MSc. Other thesis committee members Dr.ir. R.J.Fokkink Dr.ir. J.K. Ryan October, 2012 Delft Table of Contents Preface ix 1 Introduction1 1-1 Option pricing.................................... 1 1-2 Contents of the thesis................................ 4 2 Problem description5 2-1 Introduction to Fourier series and the Fourier transform.............. 5 2-1-1 Fourier series................................ 5 2-1-2 Fourier transform.............................. 7 2-1-3 Characteristic functions........................... 8 2-2 Convergence..................................... 9 2-2-1 Convergence rates.............................. 9 2-2-2 Convergence of the Fourier series...................... 10 2-3 Gibbs phenomenon................................. 11 2-3-1 Introduction................................. 11 2-3-2 Gibbs constant................................ 12 2-3-3 Gibbs phenomenon in applications..................... 13 2-4 COS method..................................... 14 2-4-1 Truncation Range.............................. 18 2-4-2 Error Analysis................................ 18 2-5 Models of interest and Gibb’s phenomenon..................... 19 2-5-1 The Black-Scholes Model.........................
    [Show full text]
  • An Adaptive Approach to Gibbs' Phenomenon
    The University of Southern Mississippi The Aquila Digital Community Master's Theses Summer 2020 An Adaptive Approach to Gibbs’ Phenomenon Jannatul Ferdous Chhoa Follow this and additional works at: https://aquila.usm.edu/masters_theses Part of the Numerical Analysis and Computation Commons, and the Signal Processing Commons Recommended Citation Chhoa, Jannatul Ferdous, "An Adaptive Approach to Gibbs’ Phenomenon" (2020). Master's Theses. 762. https://aquila.usm.edu/masters_theses/762 This Masters Thesis is brought to you for free and open access by The Aquila Digital Community. It has been accepted for inclusion in Master's Theses by an authorized administrator of The Aquila Digital Community. For more information, please contact [email protected]. AN ADAPTIVE APPROACH TO GIBBS’ PHENOMENON by Jannatul Ferdous Chhoa A Dissertation Submitted to the Graduate School, the College of Arts and Sciences and the School of Mathematics and Natural Sciences of The University of Southern Mississippi in Partial Fulfillment of the Requirements for the Degree of Master of Science Approved by: Dr. James Lambers, Committee Chair Dr. Haiyan Tian Dr. Huiqing Zhu Dr. James Lambers Dr. Bernd Schroeder Dr. Karen S. Coats Committee Chair Director of School Dean of the Graduate School August 2020 COPYRIGHT BY JANNATUL FERDOUS CHHOA 2020 ABSTRACT Gibbs’ Phenomenon, an unusual behavior of functions with sharp jumps, is encountered while applying the Fourier Transform on them. The resulting reconstructions have high frequency oscillations near the jumps making the reconstructions far from being accurate. To get rid of the unwanted oscillations, we used the Lanczos sigma factor to adjust the Fourier series and we came across three cases.
    [Show full text]
  • Gibbs Phenomenon and Its Removal for a Class of Orthogonal Expansions
    Gibbs phenomenon and its removal for a class of orthogonal expansions Ben Adcock DAMTP, Centre for Mathematical Sciences University of Cambridge Wilberforce Rd, Cambridge CB3 0WA United Kingdom August 5, 2010 Abstract We detail the Gibbs phenomenon and its resolution for the family of orthogonal ex- pansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Dirichlet boundary conditions. As we establish, it is possible to completely describe this phenomenon, including determining exact values for the size of the overshoot near both the domain boundary and the interior discontinuities of the function. Next, we demonstrate how the Gibbs phenomenon can be removed from such expansions using a number of different techniques. As a by-product, we introduce a generalisation of the classical Lidstone polynomials. 1 Introduction Fourier series lie at the heart of countless methods in computational mathematics. Unfortunately, whenever a piecewise smooth function is represented by its Fourier series, the approximation suf- fers from the well-known Gibbs phenomenon [24, 36]. Several characteristics of this phenomenon include the slow convergence of the expansion away from the discontinuity locations, the lack of uniform convergence and the presence of O (1) oscillations near discontinuities [38]. In particu- lar, the maximal overshoot of the Fourier series of a function f near any discontinuity x0 is of + − size c[f(x0 ) − f(x0 )], where c ≈ 0:0895. It is a testament to the importance of the Gibbs phenomenon that the development of techniques for its amelioration, and indeed, complete removal, remains an active area of inquiry. The list of existing methods includes filtering [36], Gegenbauer reconstruction [20, 21], techniques based on extrapolation [14, 15, 16], Pad´emethods [13] and Fourier extension/continuation meth- ods [10, 22], to name but a few (for a more comprehensive survey see [11, 36] and references therein).
    [Show full text]
  • 5: Gibbs Phenomenon Extension: Method (A) T2 Periodic Extension: Method (B) Summary
    5: Gibbs ⊲ Phenomenon Discontinuities Discontinuous Waveform Gibbs Phenomenon Integration Rate at which coefficients decrease with m Differentiation Periodic Extension t2 Periodic 5: Gibbs Phenomenon Extension: Method (a) t2 Periodic Extension: Method (b) Summary E1.10 Fourier Series and Transforms (2014-5559) Gibbs Phenomenon: 5 – 1 / 11 Discontinuities 5: Gibbs Phenomenon A function, v(t), has a discontinuity of amplitude b at t = a if ⊲ Discontinuities Discontinuous Waveform lime→0 (v(a + e) − v(a − e)) = b 6= 0 Gibbs Phenomenon Integration Rate at which Conversely, v(t), is continuous at t = a if the limit, b, equals zero. coefficients decrease with m Differentiation Periodic Extension Examples: t2 Periodic Extension: Method b b (a) u(t) v(t) t2 Periodic 0 0 Extension: Method a – e a a + e a – e a a + e (b) Time (t) Time (t) Summary Continuous Discontinuous We will see that if a periodic function, v(t), is discontinuous, then its Fourier series behaves in a strange way. E1.10 Fourier Series and Transforms (2014-5559) Gibbs Phenomenon: 5 – 2 / 11 Discontinuous Waveform 5: Gibbs Phenomenon Pulse: T = 1 = 20, width= 1 T , height A = 1 1 Discontinuities F 2 0.5 Discontinuous . T 0 ⊲ Waveform 1 0 5 −i2πmFt 0 5 10 15 20 Gibbs Phenomenon Um = Ae dt T 1 0 max(u )=0.500 Integration 0 0.5T 0.5 i −i2πmFt N=0 Rate at which = R e 0 coefficients decrease 2πmFT 0 0 5 10 15 20 with m m i i 1 −iπm ((−1) −1) max(u )=1.137 Differentiation e 1 = 2πm − 1 = 2πm 0.5 N=1 Periodic Extension 0 t2 Periodic 0 5 10 15 20 Extension: Method 0 m 6= 0, even 1 max(u )=1.100 (a) 3 0.5 2 = 0.5 m = 0 N=3 t Periodic 0 Extension: Method −i 0 5 10 15 20 (b) mπ m odd 1 max(u )=1.094 Summary 5 0.5 1 2 1 N=5 So, u(t) = + sin 2πFt + sin 6πFt 0 2 π 3 0 5 10 15 20 1 1 + sin 10πFt + ..
    [Show full text]
  • Gibbs Phenomenon in Engineering Systems
    Gibbs Phenomenon and its Applications in Science and Engineering Josué Njock Libii Engineering Department Indiana University-Purdue University Fort Wayne Fort Wayne, Indiana, 46805-1499 [email protected] Abstract Gibbs phenomenon arises in many applications. In this article, the author first discusses a brief history of this phenomenon and several of its applications in science and engineering. Then, using the Fourier series of a square-wave function and computer software in a classroom exercise, he illustrates how Gibbs phenomenon can be used to illustrate to undergraduate students the concept of nonuniform convergence of successive partial sums over the interval from 0 to B. 1. Introduction Gibbs Phenomenon is intimately related to the study of Fourier series. When a periodic function f(x) with a jump discontinuity is represented using a Fourier series, for example, it is observed that calculating values of that function using a truncated series leads to results that oscillate near the discontinuity [12]. As one includes more and more terms into the series, the oscillations persist but they move closer and closer to the discontinuity itself. Indeed, it is found that the series representation yields an overshoot at the jump, a value that is consistently larger in magnitude than that of the actual function at the jump. No matter how many terms one adds to the series, that overshoot does not disappear. Thus, partial sums that approximate f(x) do not approach f(x) uniformly over an interval that contains a point where the function is discontinuous [23]. This behavior, which appears in many practical applications, is known as Gibbs Phenomenon; it is a common example that is used to illustrate how nonuniform convergence can arise [3].
    [Show full text]
  • Applications of Gibbs Phenomenon in Engneering Education
    Session 3265 Applications of Gibbs Phenomenon in Engineering Education Josué Njock Libii Engineering Department Indiana University-Purdue University Fort Wayne Fort Wayne, Indiana, 46805-1499 1. Introduction. Nonuniform convergence arises in many engineering applications. In this article, the author demonstrates how Gibbs phenomenon associated with the Fourier series of a square-wave function and computer software can be used to illustrate the concept of nonuniform convergence of successive partial sums over the interval from 0 to π. Trigonometric series of the form ¥ a0 f (x) = + å(an cosnx + bn sin nx) 2 n=1 are important in many problems in physics, engineering and biology. Here, a0 ,an ,bn , are coefficients to be determined. Such series and periodic functions that can be represented by them arise naturally in many practical applications. Yamashiro and Grudins[20] used Fourier series to model the flow of air in the lungs; many electric sources of energy generate waveforms that are periodic; sweep generators used to control electron beams in cathode-ray oscilloscopes produce triangular waves; nonlinearities that arise in circuits that are supposed to be linear create periodic functions; and when a nonfiltered electronic rectifier is driven by a sinusoidal source, it produces sine waves that are rectified [21]. Fourier series are used in a wide variety of other fields including electromagnetic wave theory [6], heat transfer [9], acoustics [23], mechanical and structural vibrations [11], and frequency analysis of signals [5]. In contrast to Taylor series, which can only be used to represent functions that have many derivatives, trigonometric series can be used to represent functions that are continuous as well as Page 8.228.1 Page "Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition Copyright © 2003, American Society1 for Engineering Education" those that are discontinuous.
    [Show full text]
  • The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations Mathematics and Its Applications
    The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations Mathematics and Its Applications Managing Editor: M.HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands Volume 446 The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations by Abdul J. Jerri Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York, U.S.A. Springer·Science+Business Media, B.V A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4419-4800-7 ISBN 978-1-4757-2847-7 (eBook) DOI 10.1007/978-1-4757-2847-7 Softcover reprint of the hardcover 1st edition 1998 Printed on acid-free paper All Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 . No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner To my caring wife Suad with gratitude Contents Preface xiii ACKNOWLEDGEMENTS xvii AIM OF THE BOOK xix 1 INTRODUCTION 1 1.1 The Gibbs-Wilbraham Phenomenon 1 1.2 Some Basic Elements of Fourier Analysis. 3 1.3 Illustrations and Analysis . 12 A. The Truncated Fourier Series Approximation 12 B. The Truncated Fourier Integral Approximation 16 1.4 Filtering via the Fejer Averaging 26 A. The Fejer Averaging ..... 28 B. The (C, a) Summability ... 31 1.5 The Lanczos-Local-Type Filtering 34 2 ANALYSIS AND FILTERING 37 2.1 The Truncated Fourier Integral .
    [Show full text]