61. Fourier Analysis fourier analysis. In March 1984 we paid a visit to the physics department of the University of Toronto to photograph the collection of Koenig apparatus. Prof. http://physics.kenyon.edu/EarlyApparatus/Rudolf_Koenig_Apparatus/Fourier_Analysi

Fourier Analysis In March 1984 we paid a visit to the physics department of the University of Toronto to photograph the collection of Koenig apparatus. Prof. Malcolm Graham, our host, told me that this "Manometric Flame Analyser for the timbre of sounds, with 14 universal resonators ... 650 francs" ($130) had recently been put into operation and worked properly. Today we would call this a Fourier analyzer. The adjustable Helmholtz resonators (see the detail at the right, below) are tuned to the fundamental frequency of the sound to be analyzed, plus its harmonics. The holes on the other side of the resonators are connected by the rubber tubes to manometric flame capsules , and the variation in the height of the flames observed in the rotating mirror. The variation is proportional to the strength of the Fourier component of the sound. The picture at the left, below, shows the manometric capsules and the jets where the flames are produced. Note the black background to made the flames more visible. The Fourier analyzer at the right in the Garland Collection of Classic Physics Apparatus at Vanderbilt University in Nashville, Tennessee. It arrived from France in time for the opening of Vanderbilt in the fall of 1875.

62. Fourier Analysis, Links, Books Jean Baptiste Joseph Fourier , fourier analysis. Elias M. Stein, Rami Shakarchi fourier analysis An Introduction fourier analysis An Introduction. http://www.saunalahti.fi/jawap/colour/books/fourier.html

Jean Baptiste Joseph Fourier , Fourier analysis. Mathematician, born: 1768 in Auxerre, Bourgogne, France. Died: 1830 in Paris, France Fourier series, Fourier transforms, periodic functions, Applications: Heat flow, vibrating systems, waves... Books: Elias M. Stein, Rami Shakarchi Fourier Analysis: An Introduction Murray R. Spiegel, M. R. Spiegel Schaum's Outline of Fourier Analysis Eugene Hecht Schaum's Outline of Optics Eugene Hecht Optics Transnational College of Lex Tokyo, Trans Coll, Yo Sakakibara (Introduction) Who Is Fourier: A Mathematical Adventure Joseph W. Goodman Introduction To Fourier Optics Godfrey Harold Hardy, Rogosinski, Hart Hardy Fourier Series Keyword Title Author Need to translate web pages, text? SYSTRAN Links FindGraph Are you an engineer, scientist or graduate student looking for a graphing tool that will allow you to quickly analyze graphed data without having to immerse yourself into calculus and statistics books? You've just found the perfect solution. FindGraph is a comprehensive, feature-rich graphing, digitizing and curve fitting software specially created for these purposes. Take any graph or data from any source (Web or PDF document, for example) add your comments and perform any manipulations, like building approximation lines. Then print your results or export them to Excel or other database. The program comes with a straightforward interface, 200 built-in common graphing functions (like Cardioid, Devil's Curve, or Double Folium), support for Polar, Cartesian, and Parametric equations, various data input/export features and printing options.

63. Fourier Analysis Of Tides fourier analysis of the tidal record. There are three facts about sine and cosine finctions that make fourier analysis work. First fact. http://www.math.sunysb.edu/~tony/tides/analysis.html

Fourier analysis of the tidal record The Tidal Analyzer, ( Kelvin , opposite p. 304). Once the working hypothesis is established, that the astronomical tidal function for any given port is a sum of a certain number of constituents whose frequencies are known a priori then the amplitudes and phases of the constituents may be determined by Fourier analysis. To put the sum in more standard form, a constituent H cos( vt p ) will be rewritten using a standard trigonometric identity as A cos vt B sin vt (with A H cos p and B H sin p There are three facts about sine and cosine finctions that make Fourier analysis work. First fact . In the long run, the average value of any function of the form sin( vt ) or cos( vt ) must be zero. This is clear from looking at the graphs of these functions: each positive contribution to the average is exactly cancelled by a negative one. Second fact . For different speeds v and w the average value of the product cos( vt ) cos( wt goes to zero as the average is taken over longer and longer time intervals. The reason is that in the long run the times when the two functions are out of phase (so the product is negative) will cancel the contributions from the times they are in phase. Similarly for the products cos( vt ) sin( wt cos( vt ) sin( vt sin( vt ) sin( wt Third fact . The average value of the products cos( vt ) cos( vt sin( vt ) sin( vt goes to exactly 1/2 if the averages are taken over longer and longer time intervals. First of all, in each case the two factors are always in phase, in fact equal, so their product is always either the square of a positive number or the square of a negative number, or zero, but in any case never negative, so there can be no cancellation. Why is the average exactly 1/2? Since the graphs of the sine function and the cosine function are so similar, we can expect that in the long run sine-squared and cosine-squared would have

64. Wiley::Fourier Analysis Of Time Series: An Introduction, Second Edition Wiley Mathematics Statistics Statistics Special Topics fourier analysis of Time Series An Introduction, Second Edition. http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471889482.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley Statistics Special Topics Fourier Analysis of Time Series: An Introduction, Second Edition Related Subjects Combinatorics Differential Equations Related Titles Statistics Special Topics Statistical Learning Theory (Hardcover) by Vladimir N. Vapnik Biostatistics: A Methodology For the Health Sciences, 2nd Edition (Hardcover) by Gerald van Belle, Lloyd D. Fisher, Patrick J. Heagerty, Thomas S. Lumley Statistics and the Law (Paperback) by Morris H. DeGroot (Editor), Stephen E. Fienberg (Editor), Joseph B. Kadane (Editor) A Probabilistic Analysis of the Sacco and Vanzetti Evidence (Hardcover) by Joseph B. Kadane, David A. Schum Time Series Analysis: Nonstationary and Noninvertible Distribution Theory (Hardcover) by Katsuto Tanaka Fundamentals of Queueing Theory, Third Edition (Hardcover) by Donald Gross, Carl M. Harris Nonparametric Statistical Methods, 2nd Edition (Hardcover) by Myles Hollander, Douglas A. Wolfe Join a Statistics Special Topics Fourier Analysis of Time Series: An Introduction, Second Edition

65. Wiley::Fourier Analysis On Groups Wiley Mathematics Statistics Algebra Complex Functional Analysis fourier analysis on Groups. Related Subjects, http://www.wiley.com/WileyCDA/WileyTitle/productCd-047152364X.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley Algebra Fourier Analysis on Groups Related Subjects General Algebra Linear Algebra Related Titles Applied and Computational Complex Analysis, Volume 2, Special Functions-Integral Transforms- Asymptotics-Continued Fractions (Paperback) by Peter Henrici Functional Analysis (Hardcover) by Peter D. Lax Applied and Computational Complex Analysis, Volume 3, Discrete Fourier Analysis, Cauchy Integrals, Construction of Conformal Maps, Univalent Functions (Paperback) by Peter Henrici Applied and Computational Complex Analysis, 3 Volume Set (Paperback) by Peter Henrici Applied and Computational Complex Analysis, Volume 1, Power Series Integration Conformal Mapping Location of Zero (Paperback) by Peter Henrici Linear Operators, Part 3, Spectral Operators (Paperback) by Neilson Dunford, Jacob T. Schwartz Linear Operators, Part 2, Spectral Theory, Self Adjoint Operators in Hilbert Space (Paperback) by Neilson Dunford, Jacob T. Schwartz Join a Fourier Analysis on Groups Walter Rudin ISBN: 0-471-52364-X Paperback 296 pages January 1990 US $125.00

66. Studiehandbok 04/05 Course. Coordinator Se kurssida/ See course page. fourier analysis *. A course of Fourier series and Fourier integrals. Aim. To make http://www.kth.se/student/studiehandbok/Kurs.asp?Code=5B1466&Lang=1

67. Schaums Outline Of Fourier Analysis With Applications To Boundary Value Problems Schaums Outline of fourier analysis with Applications to Boundary Value Problems. Schaums Outline of fourier analysis with Applications http://www.sciencesbookreview.com/Schaums_Outline_of_Fourier_Analysis_with_Appli

Schaums Outline of Fourier Analysis with Applications to Boundary Value Problems Schaums Outline of Fourier Analysis with Applications to Boundary Value Problems by Authors: Murray R Spiegel Released: 01 March, 1974 ISBN: 0070602190 Paperback Sales Rank: List price: Our price: You save: Book > Schaums Outline of Fourier Analysis with Applications to Boundary Value Problems > Customer Reviews: Average Customer Rating: Schaums Outline of Fourier Analysis with Applications to Boundary Value Problems > Customer Review #1: Step-by-step explanations This text is a good supplement to understanding the use of Fourier analysis and how it is used in real-world applications. The explanations are to the point and the solved problems are all fairly easy to follow. At the end of the chapter, there are exercises to test your knowledge, and most of the answers are in the back of the book. Modeling the exercises on the problems, you can usually work out what you should do for the exercise. This is a good study guide.

68. U Of T PSY280F: Fourier Analysis fourier analysis. Bennett s PSY380F site; Prof. Krantz s Pictorial Introduction to fourier analysis (Hanover College); The life and times of Fourier http://www.cquest.utoronto.ca/psych/psy280f/ch5/fourier.html

Fourier Analysis some useful outside links (these have no direct links back to PSY280): Prof. Bennett's site Prof. Krantz's Pictorial Introduction to Fourier Analysis (Hanover College) The life and times of Fourier You can combine sinewave gratings point by point. Here, combining a vertical and horizontal grating, we end up with something like a checkerboard. The final pattern you get depends on the input. When you combine sinewave gratings, the result depends on the properties of each individual sinewave grating (contrast, orientation, spatial frequency) and their relative phases (positions relative to one another). In this figure, the two sinewave gratings on the top have been added together to get the two different patterns on the bottom. The reason the two bottom patterns look different is because we added the stimuli together in different relative phases in the two cases. Creating more complex patterns out of sinewave gratings works the same way as with simple patterns: according to Fourier's Theorem, a complex pattern (like a face) is just the sum of a particular set of sinewave gratings. You can think of the low spatial frequency information as containing the coarse features, and the high spatial frequency information as containing the fine details. The first image of Archie's face shows the original. The second shows what the image looks like with only low spatial frequencies. The third shows what the image looks like with only high spatial frequencies. View a movie showing how we can construct Archie's face out of sinewave gratings.

69. MESA & Fourier Analysis An outgrowth of the scientific and engineering technique of fourier analysis, MESA makes a radical break from earlier cycledetection methods. http://www.aspenres.com/Documents/help/userguide/help/Mesahelp/mesa1MESA__Fourie

MESA, which stands for Maximum Entropy Spectrum Analysis, is an advanced mathematical method for filtering any cyclical components of different frequencies from complex signals or data sets. MESA has been used to identify the cyclic components of data sets that originate from chaotic bursts of radio waves, sub-terranean explosions, and, more recently, from military radar. An outgrowth of the scientific and engineering technique of Fourier analysis, MESA makes a radical break from earlier cycle-detection methods. Traditional Fourier techniques such as the fast Fourier transform identify cycles with a high degree of certainty, but require very large data samples an integral multiple of the wavelength of the cycle or cycles detected. Furthermore, even larger data samples are required to allow good resolution identification of concurrent cycles of unrelated wavelengths. These restrictions render traditional Fourier methods impractical for real-time market applications because cycles recurring over a long period tend to be apparent on a bar chart, without the aid of complex mathematics. MESA, by contrast, focuses on the identification of the maximum amount of cyclic activity in a very short data sample. Whereas Fourier techniques work best for identifying cycles whose wavelengths are a tiny fraction of the length of the data samples, MESA can find a cycle in a sample only as long as the wavelength itself. This sensitivity equips MESA uniquely to pick out market cycles as they develop in fast-moving markets.

70. Fourier Analysis Of (4.15) next up previous Next Alternative approach to sound Up Sound Waves Basic Previous Sound Waves - Basic fourier analysis of (4.15). http://www-solar.mcs.st-andrews.ac.uk/~alan/sun_course/Chapter4/node4.html

Next: Alternative approach to sound Up: Sound Waves - Basic Previous: Sound Waves - Basic Fourier Analysis of ( Since the coefficients of ( ) are constant in both space and time we may look for plain waves and Fourier analyse by assuming where is a constant and . In ( ) we have and This form for is particularly useful since we note that Thus, defining the total wavenumber as ) becomes So either (and we have the trivial solution since and so ) or or the coefficient of in ( ) must vanish. Hence, This is the dispersion relation for sound waves and it relates the frequency with which the waves oscillates in time to the spatial length scales of the wave through the wave vector (and the various wave numbers). The dispersion relation, , can be used to define two important quantities, namely the phase speed and the group velocity The phase speed, in general, is given by and in the case of sound waves this is Thus, the phase speed is the sound speed. The group velocity is For sound waves Differentiating this gives on using ( The phase speed gives the speed of an individual wave and the phase velocity is . The group velocity gives the speed and direction of the transport of information and energy. Next: Alternative approach to sound Up: Sound Waves - Basic Previous: Sound Waves - Basic Prof. Alan Hood

71. Wiley Canada::Fourier Analysis Of Time Series: An Introduction, Second Edition Wiley Canada Mathematics Statistics Statistics Special Topics fourier analysis of Time Series An Introduction, Second Edition. http://www.wiley.ca/WileyCDA/WileyTitle/productCd-0471889482.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley Canada Statistics Special Topics Fourier Analysis of Time Series: An Introduction, Second Edition Related Subjects Combinatorics Differential Equations Related Titles Statistics Special Topics Statistical Learning Theory (Hardcover) by Vladimir N. Vapnik Biostatistics: A Methodology For the Health Sciences, 2nd Edition (Hardcover) by Gerald van Belle, Lloyd D. Fisher, Patrick J. Heagerty, Thomas S. Lumley Statistics and the Law (Paperback) by Morris H. DeGroot (Editor), Stephen E. Fienberg (Editor), Joseph B. Kadane (Editor) A Probabilistic Analysis of the Sacco and Vanzetti Evidence (Hardcover) by Joseph B. Kadane, David A. Schum Time Series Analysis: Nonstationary and Noninvertible Distribution Theory (Hardcover) by Katsuto Tanaka Fundamentals of Queueing Theory, Third Edition (Hardcover) by Donald Gross, Carl M. Harris Nonparametric Statistical Methods, 2nd Edition (Hardcover) by Myles Hollander, Douglas A. Wolfe Statistics Special Topics Fourier Analysis of Time Series: An Introduction, Second Edition

72. Wiley Canada::Fourier Analysis On Groups Wiley Canada Mathematics Statistics Algebra Complex Functional Analysis fourier analysis on Groups. Related Subjects, http://www.wiley.ca/WileyCDA/WileyTitle/productCd-047152364X.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley Canada Algebra Fourier Analysis on Groups Related Subjects General Algebra Linear Algebra Related Titles Applied and Computational Complex Analysis, Volume 2, Special Functions-Integral Transforms- Asymptotics-Continued Fractions (Paperback) by Peter Henrici Functional Analysis (Hardcover) by Peter D. Lax Applied and Computational Complex Analysis, Volume 3, Discrete Fourier Analysis, Cauchy Integrals, Construction of Conformal Maps, Univalent Functions (Paperback) by Peter Henrici Applied and Computational Complex Analysis, 3 Volume Set (Paperback) by Peter Henrici Applied and Computational Complex Analysis, Volume 1, Power Series Integration Conformal Mapping Location of Zero (Paperback) by Peter Henrici Linear Operators, Part 3, Spectral Operators (Paperback) by Neilson Dunford, Jacob T. Schwartz Linear Operators, Part 2, Spectral Theory, Self Adjoint Operators in Hilbert Space (Paperback) by Neilson Dunford, Jacob T. Schwartz Fourier Analysis on Groups Walter Rudin ISBN: 0-471-52364-X Paperback 296 pages January 1990 CDN $181.50

73. Powell's Books - Used, New, And Out Of Print Mathematics fourier analysis There are 82 books in this aisle. in detail fourier analysis, with emphasis on positivity and also on some function spaces and http://www.powells.com/subsection/MathematicsFourierAnalysis.html

Rare Books Technical Books Kids' Books eBooks ... Materials Science Mathematics Mechanical Engineering Media Metal Metaphysics ... view all sections... Fourier Analysis There are 81 books in this aisle. Browse the aisle by Title by Author by Price See recently arrived used books in this aisle. Featured Titles in Mathematics -Fourier Analysis: Page 1 of 3 next Used Trade Paper List Price $25.00 add to wish list Student's Guide To Fourier Transforms : With Applications in Physics and Engineering (2ND 02 Edition) by John F. James Synopsis An undergraduate guide to the basics of an important mathematical technique.... read more about this title check for other copies New Trade Paper add to wish list A Guide to Distribution Theory and Fourier Transforms by Robert S Strichartz Publisher Comments Readership: Graduate students and researchers in pure or applied mathematics and related disciplines.... read more about this title check for other copies Used Trade Paper List Price $33.00

74. F32SMS: Applications Of Fourier Analysis - Home Welcome to the Applications of fourier analysis (F32SMS) website. This website contains information related to the 2nd year Applications http://www.nottingham.ac.uk/~ppzpjm/F32SMS/

Applications of Fourier Analysis (F32SMS) Home/ Module Information Lecture Notes Problems classes Links ... Feedback Welcome to the Applications of Fourier Analysis (F32SMS) website. This website contains information related to the 2nd year Applications of Fourier Analysis Comments, criticisms, and suggestions for improvement should be sent to the module convenors at philip.moriarty@nottingham.ac.uk or david.rourke@nottingham.ac.uk Lord Kelvin. [Click on the image in the top left hand corner of this page for a biography of Fourier] ERRATA The definition of Parseval's theorem in the slides for Lecture 2 shouldn't have the 1/(2 PI) factor. The definition of the 2D FFT in the slides for Lecture 4 should have a 1/(2 PI) rather than a 1/sqrt(2 PI) factor Recent changes to website 7th May 2004: Solutions for Problems Class 8 posted. 6th May 2004: Set of notes for Lectures 1 - 4 uploaded. Click on 'Lecture Notes' above. 6th May 2004: Questions for Problems Class 7 and Problems Class 8 posted. Solutions for Problems Class 7 posted. Click on 'Problems Classes' above.

75. Mathematics And Its Applications The Gibbs Phenomenon in fourier analysis, Splines and Wavelet Approximations Abdul J. Jerri Clarkson University KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON http://www.clarkson.edu/~jerria/solnman/gibbs.html

76. Fourier Analysis And Number Theory fourier analysis and number theory. In this page we consider a range of material which involves fourier analysis in a number theoretical setting. http://www.maths.ex.ac.uk/~mwatkins/zeta/NTfourier.htm

Fourier analysis and number theory [reproduced from Brian Conrey's survey article on the Riemann hypothesis Notices of the AMS (2003), p. 346] The above graph illustrates a Fourier-type duality relation between the prime numbers and the zeros of the Riemann zeta function . The function depicted is the Fourier transform of the error term in the Prime Number Theorem , and the visible spikes correspond to the imaginary parts of the zeta zeros. This Fourier-type duality is most clearly expressed mathematically by the Riemann-Weil explicit formula . What is most interesting, and least understood, about this situation is the fact that the structure of the explicit formula is mirrored by certain dynamical trace formulae The first instance of this to be observed involved the Selberg trace formula (discovered in the 1950's) which concerns the geodesic flow on a Riemann surface, relating its periodic orbits and its energy levels, i.e. eigenvalues of the Laplace-Beltrami operator. Here the orbits correspond to the primes and the energy levels to the Riemann zeta zeros. The latter correspondence lends credence to the spectral interpretation of the Riemann zeta function , and the overall situation suggests the existence of some kind of mysterious dynamical system underlying (or "lurking behind" as N. Snaith put it in

77. Math 750, Fall 2002 MATH 750 fourier analysis Fall 2002 Professor Robert Sharpley Meets MWF 1115-1205 in LeConte College 316. Instructor Information http://www.math.sc.edu/~sharpley/math750/

MATH 750 Fourier Analysis - Fall 2002 Professor Robert Sharpley Meets: MWF 11:15-12:05 in LeConte College 316 Instructor Information Office: LeConte 313 D Office Hours: (TBD) Course Topics The course is the study of the basic principles of Fourier analysis and the necessary prerequisites for the analysis of wavelets. Lectures will drawn from several references (listed below) and will include the following topics: Fourier series of periodic functions and the Fourier transform on the line: representation of functions, i.e. convergence and divergence (point-wise sense, in the norms of various function spaces, and almost everywhere), convergence of Fejer means and summability; Parseval's relation and the square summable theory; conjugate Fourier series, the conjugate function and the Hilbert transform, the Hardy-Littlewood maximal operator, the Riesz-Thorin and Marcinkiewicz interpolation theorems, function spaces, Riesz' theorem. Applications will include topics in the theory of partial differential equations and signal processing, in particular the FFT. Prerequisites Real Analysis ( Math 703-704 Lectures: Link to Weekly Outline Primary References "Introduction to Fourier Analysis and Wavelets" by Mark A. Pinsky, Brooks/Cole, Oct 2001. (432 pp.)

78. MA496 Signal Processing, Fourier Analysis And Wavelets MA496, Signal Processing, fourier analysis and Wavelets, 18 CATS. Status Not in PYDC in 20032004. This course may not be available http://www.maths.warwick.ac.uk/undergrad/pydc/mauve/mauve-MA496.html

MATHEMATICS INSTITUTE A-Z Index Search Mauve (M-level) PYDC 2003-2004 Overview (White) Study Guide (Orange) Year 1 (Blue) Year 2 (Green) ... University Signal Processing, Fourier Analysis and Wavelets 18 CATS Status : Not in PYDC in 2003-2004. This course may not be available this year, or may only be available by special arrangement. Section: Choose a section Front Page Introduction MMath MSc MA4xx Modules MA5xx Modules MA6xx Modules Search PYDC: See also: Location of Lecture Rooms InSite

79. Fourier Analysis Of Neural Data fourier analysis of Neural Data. Introduction. On December 21, 1807 a young Why use fourier analysis? A ubiquitous feature of Neural Systems http://www-users.york.ac.uk/~dh20/Fourier1.html

Fourier Analysis of Neural Data Introduction. On December 21, 1807 a young French engineer addressed the eminent mathematicians of the French Academy, and made what seemed like an incredible claim. He stated that any arbitrary function, defined over a finite interval, could be represented as an infinite summation of cosine and sine functions. This claim was disputed by the members of the Academy, and it was some time before it was accepted for what it was - one of the major advances in Mathematics, now known as Fourier's theorem, named after it's originator: Jean Baptiste Joseph de Fourier (1768-1830). Why use Fourier Analysis? A ubiquitous feature of Neural Systems is the presence of Noise at all levels, from single ion channels through to fluctuations in our movements. The presence of such stochastic elements necessitates the use of a Statistical Signal Processing framework. Part of my research work has been the development of a multivariate Fourier based framework for the analysis of neural data - both spike train and time series. This framework provides a powerful rigorous, statistical framework within which to characterize linear and non-linear interactions between stochastic signals. In recent years there has been an increased use of spectral methods for the analysis of neural data, both spike train and time series data. Since the basis of this framework is the decomposition of signals into constituent frequency components - these methods are particularly suited for the analysis of rhythmic neural activity, and oscillatory neural systems.

80. An Introduction To Fourier Theory Linear transforms, especially fourier and Laplace transforms, are widely used in solving problems The fourier transform is used in linear systems analysis, antenna studies, optics http://aurora.phys.utk.edu/~forrest/papers/fourier

An Introduction to Fourier Theory by Forrest Hoffman This paper is also available in DVI , and PostScript Table of Contents Introduction The Fourier Transform The Two Domains Fourier Transform Properties ... References Introduction Linear transforms, especially Fourier and Laplace transforms, are widely used in solving problems in science and engineering. The Fourier transform is used in linear systems analysis, antenna studies, optics, random process modeling, probability theory, quantum physics, and boundary-value problems ( Brigham , 2-3) and has been very successfully applied to restoration of astronomical data ( Brault and White ). The Fourier transform, a pervasive and versatile tool, is used in many fields of science as a mathematical or physical tool to alter a problem into one that can be more easily solved. Some scientists understand Fourier theory as a physical phenomenon, not simply as a mathematical tool. In some branches of science, the Fourier transform of one function may yield another physical function ( Bracewell The Fourier Transform The Fourier transform , in essence, decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform. It identifies or distinguishes the different frequency sinusoids and their respective amplitudes (

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