Fourier Integration

9 Key excerpts on "Fourier Integration"

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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)

  It is possible to define the Fourier transform of a function of several variables, which is important for instance in the physical study of wave motion and optics. It is also possible to generalize the Fourier transform on discrete structures such as finite groups. The efficient computation of such structures, by fast Fourier transform, is essential for high-speed computing. Definition There are several common conventions for defining the Fourier transform of an integrable function ƒ : R → C (Kaiser 1994). Here we will use the definition: for every real number ξ . When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, ƒ can be recon-structed from by the inverse transform : for every real number x . For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ , see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. ________________________ WORLD TECHNOLOGIES ________________________ Introduction The motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series, complicated functions are written as the sum of simple waves mathematically represented by sines and cosines. Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use Euler's formula, which states that e 2 πiθ = cos 2 πθ + i sin 2 πθ , to write Fourier series in terms of the basic waves e 2 πiθ . This has the advantage of simplifying many of the formulas involved and providing a formulation for Fourier series that more closely resembles the definition followed here. This passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex valued.

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  Digital Signal Processing

  Mathematical and Computational Methods, Software Development and Applications

  • Jonathan M Blackledge(Author)
  • 2006(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Some years later, Fourier developed his transform to investigate solutions to this equation; a transform that was essentially invented in an attempt solve an equation derived from a military problem of the time. Since then, the Fourier transform has found applications in nearly all areas of science and engineering and is arguably one of the most, if not, the most important integral transforms ever devised.

  4.1.1 Notation

  The Fourier transform of a function f is usually denoted by the upper case F but many authors prefer to use a tilde above this function, i.e. to denote the Fourier transform of f by

  f ˜

  . In this work, the former notation is used throughout. Thus, the Fourier transform of f can be written in the form

  F ω ≡

  F ^

  1

  f t =

  ∫

  − ∞

  ∞

  f t exp

  − iωt

  dt

  where

  F ^

  1

  denotes the one-dimensional Fourier transform operator. Here, F (ω ) is referred to as the Fourier transform of f (t ) where f (t ) is a non-periodic function (see Chapter 3 ).

  The sufficient condition for the existence of the Fourier transform is that f is square integrable, i.e.

  ∫

  − ∞

  ∞

  f t

  2

  dt

  < ∞ .

  4.1.2 Physical Interpretation

  Physically, the Fourier transform of a function provides a quantitative picture of the frequency content of the function which is important in a wide range of physical problems and is fundamental to the processing and analysis of signals and images. The variable ω has dimensions that are reciprocal to those of the variable t. There are two important cases which arise:

  (i)  

  t is time in seconds and ω is the temporal frequency in cycles per second (Hertz). Here, ω is referred to as the angular frequency which is given by 2π × v where v is the frequency.

  (ii)  

  t is distance in metres (usually denoted by x ) and ω and the spatial frequency in cycles per metre (usually denoted by κ). Here, κ is known as the wavenumber and is given by

  k =

  2 π

  λ

  where λ is the wavelength and we note that

  c =

  ω k

  = v λ

  where c

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  Numerical and Analytical Methods with MATLAB for Electrical Engineers

  • William Bober, Andrew Stevens(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  239 Chapter 8 Fourier Transforms and Signal Processing 8.1 Introduction In electronic circuits, we are often presented with voltage and currents that repre-sent signals that carry meaningful information. Examples include ◾ audio, including hi-fi and voice ◾ video, including television and surveillance ◾ telemetry from temperature sensors, strain gauges, medical instrumentation, power meters, and the like ◾ radar, loran, and GPS signals for detection and guidance systems on aircraft and ships ◾ modulated signals as AM, FM, VHF, UHF, CDMA (code division multiple access), GSM (Groupe spéciale mobile), and Wi-Fi for broadcast and point-to-point transmission of audio, video, voice, and data There are countless reasons for processing signals. For audio and video, we might want to filter out noise or reduce bandwidth. For telemetry signals, we may want to examine the data from multiple sensors to achieve a goal such as predicting the 240 ◾ Numerical and Analytical Methods with MATLAB weather, reducing an automobile’s fuel consumption, or interpreting sensor data from an electrocardiogram. In the case of communication systems, we can use modulation to multiplex audio or video signals (from radio or TV stations) onto high-frequency carriers, thereby allowing multiple signals to be carried over a common medium (e.g., a coaxial cable or optical fiber) or through the airwaves. In addition, we typically want to account for the link loss and intersymbol interference (ISI) introduced by the channel. The primary mathematical tool used in signal processing is the Fourier trans-form . This transform provides a method for describing a signal in terms of its frequency components (i.e., in the “frequency domain”) instead of in the more familiar and tangible “time domain.” Table 8.1 lists several types of signals and their corresponding frequency ranges (i.e., the range of frequency components that might be present in the waveforms).

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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)

  222 Signals and Systems: A Primer with MATLAB® control engineer might be to design a speed regulator for a disk drive head A thor-ough understanding of control systems techniques is essential to the electrical engi-neer and is of great value for designing control systems to perform the desired task 5.1 INTRODUCTION In Chapter 4, we saw that Fourier series enabled us to represent a periodic signal as a lin-ear combination of infinite sinusoids The Fourier series expansion reveals the frequency content of the periodic signal It is also possible to analyze the frequency content of non-periodic signals The tool that enables us to do this is the Fourier transform (also known as Fourier integral) The transform assumes that a nonperiodic (or aperiodic) signal is a periodic signal with an infinite period The Fourier transform is important in analyzing and processing signals in science and engineering, especially in medical images, com-puterized axis tomography (CAT), and magnetic resonance imaging (MRI) The Fourier transform is similar to Laplace transform Both are integral trans-forms The Fourier transform may be regarded a special case of Laplace transform with s = j ω , when the Laplace transform exists Just like the bilateral Laplace trans-form, Fourier transform can deal with systems having inputs for t < 0 as well as those for t >0 Fourier transform is very useful in linear systems, filtering, communication systems, and digital signal processing, in situations where Laplace transform does not exist The chapter begins by defining the Fourier transform from Fourier series Then we present and prove the properties of Fourier transform, which are useful in deriv-ing Fourier transform pairs We discuss Parseval’s theorem, compare the Laplace and Fourier transforms, and see how the Fourier transform is applied in circuit anal-ysis, amplitude modulation, and sampling We finally

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  An Introduction to Digital Signal Processing

  • John H. Karl(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Thus motivated, we integrate Eq. (7.2) over these limits with the appropriate Fourier kernel: T/2 CT/2 f(t)e-imoiQt dt = ^a n e i { n ' m ) <** dt .-772 n J-T/2 Direct elementary integration shows that rr/2 dt = T8 nm (7.3) -T/2 Continuous Time, Discrete Frequency: The Fourier Series 129 giving r /2 f(t)e-ima,ot dt=Ta„ •T /2 This result, combined with Eq. (7.2), provides the desired pair of trans-formations, relating continuous-time periodic functions with their equally spaced discrete spectra: r T/2 f(t)e-in ot sdt (7.4a) 1 a n = j J -r / 2 /(')= X a n e in ^ (7.4b) Equations (7.4), called the Fourier series, play a major role in con-tinuous theory because periodic, continuous-time signals are so common-place in many applications. Equation (7.4a) is sometimes called the analysis equation because it separates the periodic time signal into its component line spectra, a n . Equation (7.4b) represents the Fourier synthe-sis of a periodic, continuous-time signal from the superposition of complex sinusoids with frequencies that are multiples of the fundamental. The sum may or may not extend to infinity. The a n are called Fourier coefficients. Three common examples of Fourier series, the square wave, the triangle wave, and the full wave rectified sine wave are shown in Fig. 7.1. It is interesting to note that the Fourier series, which has dominated Fourier theory since its inception, takes on only a very minor part in TIME FREQUENCY -T/2 0 T/2 -T/2 0 T/2 -T/2 0 T/2 Figure 7.1 Fourier series components of three common continuous repetitive waveforms. All three have spectral components extending to infinitely high frequencies. 130 7/ Application of the Fourier Transform modern digital signal processing. In fact, we introduce it here mostly to complete the discussion of the four types of Fourier transforms aris-ing from the four possible combinations of discrete/continuous and time/frequency.

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  Experimentation in Mathematics

  Computational Paths to Discovery

  • Jonathan M. Borwein, David H. Bailey, Roland Girgensohn(Authors)
  • 2004(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  2 Fourier Series and Integrals Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by im-portant and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal ele-ment of the theory of heat by fine experiments . . . they would have the right to judge my work and to correct it. I would submit with much pleasure . . . But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication. John Herivel, Joseph Fourier: The Man and the Physicist, 1992 I t is often useful to decompose a given function into components, ana-lyze them, and then reassemble the function again, possibly in a different way. One classical and mathematically very interesting method is to use trigonometric functions. This is the basis for the theory of Fourier analysis. One can think of a sound (a certain tone played on the violin, say) as consisting of countably many oscillations w i t h different discrete frequencies, which together define the pitch and the specific timbre of the tone. These component frequencies can be identified via Fourier analysis, i n particular by computing the Fourier series of a periodic function. Of course, in reality no oscillation is precisely periodic, and a sound w i l l consist of a continuum of frequencies. Mathematically, this is analyzed by taking the continuous Fourier transform. Thus, the Fourier transform arises from Fourier series by taking more and more frequencies into account, a process described by the Poisson summation formula. Finally, the question arises as to whether a function thus analyzed can be reconstructed from its Fourier series.

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  Applied Mathematical Methods for Chemical Engineers

  • Norman W. Loney(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  170 Applied Mathematical Methods for Chemical Engineers The form given by Equation 5�59 is particularly useful, in that if ∫ ω = -ω -∞ ∞ ( ) ( ) e d i F f u u u (5�62) then ∫ = ω ω ω -∞ ∞ ( ) 1 2π ( ) e d i f x F x (5�63) Equations 5�62 and 5�63 are the Fourier transform pair , where F ( ω ) is the Fourier transform of f ( x ) and Equation 5�63 is the inverse transform� The customary notation for the Fourier transform is F { f ( x )} and its inverse is denoted by F −1 { F ( ω )}� These nota-tions will be used in this book� Again, if f ( x ) is an odd function, then Equation 5�60 can be reinterpreted as ∫ ω = ω ∞ ( ) ( )si n d s 0 F f u u u (5�64) with inverse ∫ = ω ω ω ∞ ( ) 2 π ( )si n d s 0 f x F x (5�65) Here, F s ( ω ) is the Fourier sine transform of f ( x )� If f ( x ) is an even function, Equation 5�60 can be reinterpreted as ∫ ω = ω ∞ ( ) ( )co s d 0 F f u u u c (5�66) with inverse transform given by ∫ = ω ω ω ∞ ( ) 2 π ( )co s d c 0 f x F x (5�67) where F c ( ω ) is the Fourier cosine transform of f ( x )� Similar to Laplace transforms, there is a convolution theorem for Fourier transforms [2],[4],[7–9], which states that the Fourier transform of the convolution of two functions f ( x ) and g ( x ) is equal to the product of their Fourier transforms� That is, = { * } { ( )} { ( )} F f g F f x F g x (5�68) and the convolution obeys the commutative, associative, and distributive laws of alge-bra� The convolution of the functions f ( x ) and g ( x ) is defined to be ∫ = --∞ ∞ * ( ) ( ) d f g f u g x u u (5�69) Example 5.13 Solve for f ( x ) in the integral equation given by ∫ α = -α ≤ α ≤ α >      ∞ ( )sin d 1 , 0 1 0, 1 0 f x x x

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  Essential Mathematics for NMR and MRI Spectroscopists

  • Keith C Brown(Author)
  • 2020(Publication Date)
  • Royal Society of Chemistry

    (Publisher)

  8 The Fourier Transform

  The whole is the sum of the parts. Euclid

  8.1 Fourier Series†

  As NMR/MRI spectroscopists working with modern digital equipment we routinely acquire data in the form of the free induction decay. This is the digitised form of the analogue time domain data passing through the receiver. We wish, of course to look at these data in the frequency domain and also routinely transform the time domain data to frequency domain data for display and further analysis (Figure 8.1 ).

  Figure 8.1 The Fourier transform of time-domain data to frequency-domain data.

  This transform of the data is accomplished using the ideas of Jean Baptiste Joseph Fourier (and others). Fourier led a very interesting life to say the least. He was a young man at the time of the French revolution‡ and was very nearly beheaded during the reign of terror led by Robespierre. Eventually, he became acquainted with Napoleon and travelled with him and the French army to Egypt in 1798. Later, Napoleon made him prefect of Grenoble and during this time he met and mentored Jean-Francois Champollion, the person most responsible for the translation of the Rosetta stone,§ which was found in Egypt by Napoleon's army. It was Fourier's studies of heat conduction that brought forward what we now call Fourier analysis and it is this work that he is most associated with.

  Fourier's basic idea was that the application of infinite sums of sine and cosine functions multiplied by suitable constants can be used to represent any periodic function. To understand what is happening in our spectrometer software to accomplish this remarkable feat we must delve into the mysteries of the Fourier transform.

  Our analysis of vectors showed us that we could use unit basis vectors multiplied by suitable factors to build up an overall vector that spans the vector space. This is particularly easy to visualise in two- or three-dimensional space. Mathematically, we represent this as:

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