Fourier Series

10 Key excerpts on "Fourier Series"

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  Advanced Engineering Mathematics and Analysis

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Fourier Series and Fourier Transform Fourier Series The first four Fourier Series approximations for a square wave In mathematics, a Fourier Series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier Series is a branch of Fourier ________________________ WORLD TECHNOLOGIES ________________________ analysis. Fourier Series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier Series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier Series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. Fourier Series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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  Advanced Mathematical Analysis and Engineering Mathematics

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Fourier Series and Fourier Transform Fourier Series The first four Fourier Series approximations for a square wave In mathematics, a Fourier Series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier Series is a branch of Fourier ________________________ WORLD TECHNOLOGIES ________________________ analysis. Fourier Series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigen-solutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier Series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier Series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. Fourier Series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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  Signals and Systems

  A Primer with MATLAB

  • Matthew N. O. Sadiku, Warsame Hassan Ali(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  time-domain representation, where they are described by linear differential equations It is for these reasons that Fourier analysis is used extensively today in science and engineering This chapter begins with the trigonometric Fourier Series and how to determine the coefficients of the series Then, we consider the complex exponential Fourier Series We will discuss some properties of Fourier Series We will cover circuit analy-sis, filtering, and spectrum analyzers as engineering applications of Fourier Series We will finally see how we can use MATLAB ® to plot line spectra 4.2 TRIGONOMETRIC Fourier Series The Fourier Series can be represented in three ways, the sine–cosine, amplitude– phase, and complex exponential A periodic signal is one that repeats itself every T s In other words, a continuous time signal x ( t ) satisfies x t x t nT ( ) ( ) = + (41) where n is an integer T is the fundamental period of x ( t )

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  Bird's Higher Engineering Mathematics

  • John Bird(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  Section K Fourier Series Chapter 60 Fourier Series for periodic functions of period 2π Why it is important to understand: Fourier Series for periodic functions of periodic 2π A Fourier Series changes a periodic function into an infinite expansion of a function in terms of sines and cosines. In engineering and physics, expanding functions in terms of sines and cosines is useful because it makes it possible to more easily manipulate functions that are just too difficult to represent analytically. The fields of electronics, quantum mechanics and electrodynamics all make great use of Fourier Series. The Fourier Series has become one of the most widely used and useful mathematical tools available to any scientist. This chapter introduces and explains Fourier Series. At the end of this chapter, you should be able to: • describe a Fourier Series • understand periodic functions • state the formula for a Fourier Series and Fourier coefficients • obtain Fourier Series for given functions 664 Section K 60.1 Introduction Fourier * series provides a method of analysing peri- odic functions into their constituent components. Alter- nating currents and voltages, displacement, velocity and acceleration of slider-crank mechanisms and acoustic waves are typical practical examples in engineering and science where periodic functions are involved and often require analysis. 60.2 Periodic functions As stated in Chapter 16, a function f (x) is said to be periodic if f (x + T) = f (x) for all values of x, where T is some positive number. T is the interval between two successive repetitions and is called the period of the function f (x). For example, y = sin x is peri- odic in x with period 2π since sin x = sin(x + 2π) = sin(x + 4π), and so on. In general, if y = sin ωt then the period of the waveform is 2π/ω.

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  Advanced Mathematical Techniques in Engineering Sciences

  • Mangey Ram, J. Paulo Davim, Mangey Ram, J. Paulo Davim(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  chapter two

  Fourier Series and itsapplications in engineering

  Smita Sonker and Alka Munjal National Institute of Technology Kurukshetra Contents 2.1   Introduction 2.2   Periodic functions 2.3   Orthogonality of sine and cosine functions 2.4   Fourier Series 2.5   Dirichlet’s theorem 2.6   Riemann–Lebesgue lemma 2.7   Term-wise differentiation 2.8   Convergence of Fourier Series 2.9   Small order 2.10 Big “oh” for functions 2.11 Fourier analysis and Fourier transform 2.12 Fourier transform 2.13 Gibbs phenomenon 2.13.1 Gibbs phenomenon with an example 2.13.2 Results related to Gibbs phenomenon 2.14 Trigonometric Fourier approximation 2.15 Summability 2.15.1 Ordinary summability 2.15.2 Absolute summability 2.15.3 Strong summability 2.16 Methods for summability 2.17 Regularity condition 2.18 Norm 2.19 Modulus of continuity 2.20 Lipschitz condition 2.21 Various Lipschitz classes 2.22 Degree of approximation 2.23 Fourier Series and music 2.24 Applications and significant uses References

  2.1   Introduction

  Mathematics has its roots embedded within various streams of engineering and sciences. The concepts of the famous Fourier Series were originated from the field of physics. The following two physical problems are the reasons for the origin of Fourier Series:

  1. Heat conduction in solid
  2. The motion of a vibrating string

  Jean Baptiste Joseph Fourier (1768–1830) was the first physicist, mathematician, and engineer who developed the concepts of Fourier analysis in dealing with the problems of vibrations and heat transfer. He claimed that any continuous or discontinuous function of t could also be expressed as a linear combination of cos(t ) and sin(t ) functions.

  In the mathematical analysis, we do not usually get a full decomposition into the simpler things, but an approximation of a complex system is usually achieved by a more elementary system. When we truncate the Taylor series expansion of a function, we approximate the function by using the polynomial.

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  Fast Fourier Transforms

  • James S. Walker(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  1 Basic Aspects of Fourier Series

  This chapter is a summary of the basic theory of Fourier Series. Some of the deeper theorems are only quoted; their proofs can be found in [Wa], listed in this book’s bibliography. Besides their importance in applications, Fourier Series provide a foundation for understanding the FFT.

  1.1  Definition of Fourier Series

  To understand the definition of Fourier Series we will begin with the essential idea: representing a wave form in terms of frequency as opposed to time. We will denote time by x rather than t.

  Suppose our wave form is described by 4 cos 2π vx, which has frequency v. Using Euler’s identity

  e i ϕ = cos   ϕ + i   sin   ϕ

  (1.1)

  we can write 4 cos 2π vx in complex exponential form

  4   cos   2 π ν x = 2

  e

  i 2 π ν x

  + 2

  e

  − i 2 π ν x

  where the complex exponentials have amplitudes of 2 and frequencies of v and −v. See Figure 1.1. (To see how Figure 1.1(b) was produced, consult Exercise 1.3.)

  Or, suppose our wave form is described by 6 sin 2π vx, which also has frequency v. Using Euler’s identity (1.1 ) again, we obtain

  6   sin   2 π ν x = 3 i

  e

  − i 2 π ν x

  − 3 i

  e

  i 2 π ν x

  where the complex exponentials have complex amplitudes of 3i and −3i and frequencies of −v and v. See Figure 1.2 . (To see how Figure 1.2(b) was produced, consult Exercise 1.4.)

  FIGURE 1.1Frequency representation of 4 cos(2π vx), v = 9. (a) Graph in x domain (time or space), (b) Graph in frequency domain.

  FIGURE 1.2Frequency representation of 6 sin(2π vx), v = 9. (a) Graph in x domain (time or space), (b) Graph in frequency domain.

  These two examples show how the waves cos 2π vx and sin 2π vx can be expressed in frequency terms and distinguished from each other using complex exponentials. Also, Figures 1.1(b) and 1.2(b) show how, in a certain sense, the frequency representation of these waves is simpler.

  The basic idea in Fourier Series is to express a periodic wave as a sum of complex exponentials all of which have the same period. This is made feasible by the following property of complex exponentials having the same period.

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  Fourier and Laplace Transforms

  • R. J. Beerends, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie(Authors)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  CHAPTER 5 Applications of Fourier Series I N T R O D U C T I O N Applications of Fourier Series can be found in numerous places in the natural sci-ences as well as in mathematics itself. In this chapter we confine ourselves to two kinds of applications, to be treated in sections 5.1 and 5.2. Section 5.1 explains how Fourier Series can be used to determine the response of a linear time-invariant sys-tem to a periodic input. In section 5.2 we discuss the applications of Fourier Series in solving partial differential equations, which often occur when physical processes, such as heat conduction or a vibrating string, are described mathematically. The frequency response, introduced in chapter 1 using the response to the peri-odic time-harmonic signal e i ω t with frequency ω , plays a central role in the cal-culation of the response of a linear time-invariant system to an arbitrary periodic signal. Specifically, a Fourier Series shows how a periodic signal can be written as a superposition of time-harmonic signals with frequencies being an integer multiple of the fundamental frequency. By applying the so-called superposition rule for lin-ear time-invariant systems, one can then easily find the Fourier Series of the output. This is because the sequence of Fourier coefficients, or the line spectrum, of the out-put arises from the line spectrum of the input by a multiplication by the frequency response at the integer multiples of the fundamental frequency. For stable systems which can be described by ordinary differential equations, which is almost any linear time-invariant system occurring in practice, we will see that the frequency response can easily be derived from the differential equation. The characteristic polynomial of the differential equation of a stable system has no zeros on the imaginary axis, and hence there are no periodic eigenfunctions. As a consequence, the response to a periodic signal is uniquely determined by the differential equation.

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  Harmonic Analysis and Applications

  • John J. Benedetto(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  3 Fourier Series 3.1.  Fourier Series—Definitions and Convergence 3.1.1 Definition. Fourier Series a.  Let Ω > 0, and let F : ℝ ^ → ℂ be a function. F is 2Ω- periodic with period 2Ω if F (γ + 2Ω) = F (γ) for all γ ∈ ℝ ^. For example, F (γ) = sin y is 2π -periodic. If F is defined a.e., then F is 2Ω- periodic if F (γ + 2Ω) = F (γ) a.e. b.  Let F ∈ L loc l (ℝ ^) be 2Ω-Periodic. The Fourier Series of F is the series S (F) (γ) = ∑ ​ f [ n ] e − π i n γ / Ω, (3.1.1) where ∀ n ∈ ℤ, f [ n ] = 1 2 Ω ∫ − Ω Ω F (γ) e π i n γ / Ω d γ. (3.1.2) The numbers f [ n ] are the Fourier coefficients of F. The symbol “Σ” denotes summation over all of ℤ, i.e., “∑ n = − ∞ ∞”. c.  Formally, the right side of (3.1.1) can be thought of as defining the Fourier transform f ^ or F of the sequence f = { f [ n ]}, cf., Remark 3.1.3 d. In fact, a sequence f = { f [ n ]} is a function f : ℤ → ℂ n ↦ f [ n ]. Letting ℓ 1 (ℤ) be the space of all sequences f = { f [ n ]} for which ‖ f ‖ ℓ 1 (ℤ) ≡ ∑ ​ | f [ n ] | < ∞, the right side of (3.1.1) is well defined for f ∈ ℓ 1 (ℤ). In this context, we shall write f ↔ F, f ^ = F, f = F ˇ, (3.1.3) just as we did in Definition 1.1.2 for the case of Fourier transforms. Thus, in the case of sequences we write F ˇ [ n ] = f [ n ]. The notation (3.1.3) is based on the presumption that S (F) should equal F, e.g., Remark 3.1.2 and Theorem 3.1.6; and that if the right side of (3.1.1) defines the Fourier transform of the sequence f, then (3.1.2) is the Fourier inversion formula on ℤ corresponding to the Fourier inversion formula (1.1.1) on ℝ. In fact, S (F) often does equal F in the sense that the partial sums of the series S (F) will converge in some way to F

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  Mathematical Methods in Physics

  Partial Differential Equations, Fourier Series, and Special Functions

  • Victor Henner, Tatyana Belozerova, Kyle Forinash(Authors)
  • 2009(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  46 1. Fourier Series is a linear combination the first n functions from the infinite set {ϕ n (x)} (usually the contribution of the states decreases when n → ∞). The goal is to find the combination in Equation (1.87) that gives the best total square deviation; that is, we want to minimize the value Δ n = b integraldisplay a bracketleftbig f (x) − σ n (x) bracketrightbig 2 dx. (1.88) With Equation (1.87) we have Δ n = b integraldisplay a f 2 (x)dx − 2 n summationdisplay m =0 γ m b integraldisplay a f (x)ϕ m (x)dx + n summationdisplay m =0 γ 2 m b integraldisplay a ϕ 2 m (x)dx + 2 summationdisplay k

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  Communications Engineering

  Essentials for Computer Scientists and Electrical Engineers

  • Richard Chia Tung Lee, Mao-Ching Chiu, Jung-Shan Lin(Authors)
  • 2008(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  From Equation (3-64), we can have the following: x ð t Þ¼ x 0 þ X 1 k ¼ 1 r k cos ð 2 p kf 0 t y k Þ : ð 3-84 Þ From this we can see that Equation (3-84) is identical to Equation (3-40). This is why, although the complex exponential form Fourier Series contains imaginary terms, the imaginary terms will finally disappear because X k is a conjugate of X k . Again, as in the case where the Fourier Series is in trigonometric form, there are special cases for the exponential form. Case 1: y k ¼ 0. This occurs when b k ¼ 0. In this case, there is a cosine function with frequency kf 0 without any phase shift. Case 2: y k ¼ p 2 . This occurs when a k ¼ 0. In this case, the cosine function with frequency kf 0 becomes a sine function with frequency kf 0 without any phase shift. We may claim the following: If X k contains only a real part, the Fourier Series contains cosine functions only. If X k contains only an imaginary part, the Fourier Series contains only sine functions. Here the phase shift is p = 2 . If X k contains both real and imaginary parts, the Fourier Series contains both cosine and sine functions, or equivalently, cosine functions with phase shifts. 3.4 The Fourier Transform In contrast to the Fourier Series representation for periodic signals, the Fourier trans-form is used to represent a continuous-time nonperiodic signal as a superposition of complex sinusoids. The idea that a periodic signal can be represented as a sum of sines and cosines with Fourier Series is very powerful. We would like to extend this result to aperiodic signals. Extension of the Fourier Series to aperiodic signals can be done by extending the period to infinity. In order to take this approach, we assume that the Fourier Series of periodic extension of the nonperiodic signal x ð t Þ exists. The nonperiodic signal x ð t Þ is defined in the interval t 0 t t 0 þ T with T > 0.

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