Physics
Fourier Analysis Waves
Fourier analysis is a mathematical technique used to break down complex waveforms into simpler components, making it easier to analyze and understand their behavior. It involves expressing a function as a sum of sine and cosine functions with different frequencies and amplitudes. This approach is widely used in physics to study the behavior of waves and signals.
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10 Key excerpts on "Fourier Analysis Waves"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 10 Fourier Analysis and Fourier Transform Fourier analysis In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions. Fourier analysis is named after Joseph Fourier, who showed that representing a function by a trigonometric series greatly simplifies the study of heat propagation. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis . In mathematics, the term Fourier analysis often refers to the study of both operations. The decomposition process itself is called a Fourier transform. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis has a corresponding inverse transform that can be used for synthesis. Applications Fourier analysis has many scientific applications — in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, optics, diffraction, geometry, and other areas.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Fourier Analysis In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be repress-ented by sums of simpler trigonometric functions. Fourier analysis is named after Joseph Fourier, who showed that representing a function by a trigonometric series greatly simplifies the study of heat propagation. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis . In mathematics, the term Fourier analysis often refers to the study of both operations. The decomposition process itself is called a Fourier transform. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis has a corresponding inverse transform that can be used for synthesis. Applications Fourier analysis has many scientific applications — in physics, partial differential equa-tions, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, optics, diffraction, geometry, and other areas.
eBook - PDFIntroduction to Applied Statistical Signal Analysis
Guide to Biomedical and Electrical Engineering Applications
- Richard Shiavi(Author)
- 2010(Publication Date)
- Academic Press(Publisher)
3 FOURIER ANALYSIS 3.1 INTRODUCTION K nowledge of the cyclic or oscillating activity in various physical and biological phenomena and in engineering systems has been recognized as essential information for many decades. In fact, interest in determining sinusoidal components in measured data through modeling began at least several centuries ago and was known as harmonic decomposition . The formal development of Fourier analysis dates back to the beginning of the eighteenth century and the creative work of Jean Fourier who first developed the mathematical theory that enabled the determination of the frequency composition of mathematically expressible waveforms. This theory is called the Fourier transform and has become widely used in engineering and science. During the middle of the nineteenth century, numerical techniques were developed to determine the harmonic content of measured signals. Bloomfield (1976) summarizes a short history of these developments. One of the older phenomena that has been studied are the changes in the intensity of light emitted from a variable star. A portion of such a signal is plotted in Figure 3.1. It is oscillatory with a period of approxi-mately 25 days. Astronomers theorized that knowledge of the frequency content of this light variation could yield not only general astronomical knowledge but also information about the star’s creation (Whittaker and Robinson, 1967). Another phenomena that is usually recognized as oscillatory in nature is vibration. Accelerometers are used to measure the intensity and the frequencies of vibration, and testing them is nec-essary in order to determine their accuracy. Figure 3.2 shows the electrical output of an accelerometer that has been perturbed sinusoidally (Licht et al., 1987). Examination of the waveform reveals that the response is not a pure sinusoid. Fourier analysis will indicate what other frequency components, in addition to the perturbation frequency, are present.
eBook - PDFThe Haskell School of Music
From Signals to Symphonies
- Paul Hudak, Donya Quick(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
20 Spectrum Analysis There are many situations where it is desirable to take an existing sound signal – in particular one that is recorded by a microphone – and analyze it for its spectral content. If one can do this effectively, it is then possible (at least in theory) to recreate the original sound, or to create novel variations of it. The theory behind this approach is based on Fourier’s theorem, which states that any periodic signal can be decomposed into a weighted sum of (a potentially infinite number of) sine waves. In this chapter, we discuss the theory as well as the pragmatics for doing spectrum analysis. 20.1 Fourier’s Theorem A periodic signal is a signal that repeats itself infinitely often. Mathematically, a signal x is periodic if there exists a real number T such that for all integers n: x(t) = x(t + nT ) T is called the period, which can be just a few microseconds, a few seconds, or perhaps days – the only thing that matters is that the signal repeats itself. Usually we want to find the smallest value of T that satisfies the above property. For example, a sine wave is surely periodic; indeed, recall from Section 18.1.1 that: sin(2π k + θ) = sin θ for any integer k. In this case, T = 2π , and it is the smallest value that satisfies this property. But in what sense is, for example, a single musical note periodic? Indeed it is not, unless it is repeated infinitely often, which would not be very interesting musically. Yet something we would like to know is the spectral content of 299 300 20 Spectrum Analysis that single note, or even of a small portion of that note, within an entire composition. This is one of the practical problems that we will address later in the chapter. Recall from Section 18.1.1 that a sine wave can be represented by x(t) = A sin(ωt + φ), where A is the amplitude, ω is the radian frequency, and φ is the phase angle. Joseph Fourier, a french mathematician and physicist, showed the following result.
eBook - PDF- Sumner P. Davis, Mark C. Abrams, James W. Brault(Authors)
- 2001(Publication Date)
- Academic Press(Publisher)
4 FOURIER ANALYSIS At the outset of our discussion, we want to say that our points of view as physicists are not identical with those of mathematicians, especially with regard to functions that can or cannot be sampled, infinity, zeroes, and ill-posed problems. All of our computations are subject to the limitations of digital techniques and computers, where every function is digitally represented by a set of sampling points. Truly aperiodic functions do not exist. Infinite limits are too far away to be reached in practice, and negative frequencies do not exist. Zero means negligible, or impossible to pick out of the ubiquitous noise, simply not important enough to be considered. We do our best, within practical limits, to get the most accurate possible representation of the spectrum. Sampling allows us to treat Fourier analysis computationally but at the same time introduces other complications and artifacts that place restrictions on inter-pretation of the transforms. These restrictions will be discussed at the appropriate places in the text. In practice, Fourier analysis has two facets, the construction of a function from sinusoids (Fourier synthesis), and the decomposition of a function into its constituent sinusoids (Fourier decomposition). The two facets are reflected in the mathematical form of the Fourier integral. A function f{x) satisfying certain mathematical conditions of continuity can be represented as a superposition of sine and cosine functions f{x)^ F{a)e+''''da = F{a), (4.1) 41 42 4. Fourier Analysis where the function F{a) is termed the Fourier transform of /(x), and can itself be expressed in a similar fashion as an integral superposition of sines and cosines: / CO /(x)e-'2--dx = /(x), (4.2) -CO 4.1 Linear Systems and Superposition The essential building block in signal analysis is the concept of a linear input-output system, a black box that can be studied by examining the output response for a variety of input excitations.
No longer available |Learn more- (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.
eBook - PDF- Simon Haykin(Author)
- 2013(Publication Date)
- Wiley(Publisher)
13 CHAPTER 2 Fourier Analysis of Signals and Systems 2.1 Introduction The study of communication systems involves: • the processing of a modulated message signal generated at the transmitter output so as to facilitate its transportation across a physical channel and • subsequent processing of the received signal in the receiver so as to deliver an estimate of the original message signal to a user at the receiver output. In this study, the representation of signals and systems features prominently. More specifically, the Fourier transform plays a key role in this representation. The Fourier transform provides the mathematical link between the time-domain representation (i.e., waveform) of a signal and its frequency-domain description (i.e., spectrum). Most importantly, we can go back and forth between these two descriptions of the signal with no loss of information. Indeed, we may invoke a similar transformation in the representation of linear systems. In this latter case, the time-domain and frequency- domain descriptions of a linear time-invariant system are defined in terms of its impulse response and frequency response, respectively. In light of this background, it is in order that we begin a mathematical study of communication systems by presenting a review of Fourier analysis. This review, in turn, paves the way for the formulation of simplified representations of band-pass signals and systems to which we resort in subsequent chapters. We begin the study by developing the transition from the Fourier series representation of a periodic signal to the Fourier transform representation of a nonperiodic signal; this we do in the next two sections. 2.2 The Fourier Series Let denote a periodic signal, where the subscript T 0 denotes the duration of periodicity.
eBook - PDFIntroduction to the Senses
From Biology to Computer Science
- Terry R. J. Bossomaier(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
3.5 Fourier Analysis and the Fourier Transform Finally we get to the details of how to analyse a signal into its frequency components, or more generally from one domain to another, reciprocal domain, usually frequency. Fourier Analysis and the Fourier Transform do essentially the same thing. In the first case the domain is finite, akin to the strings on a violin and the frequencies occur in discrete, finite steps. In the latter, the domain is infinite and frequency is a continuous variable. Closely related to the continuous transform is the Discrete Fourier Transform. Most computational work in audio or image processing uses this discrete version. However, the ideas which are important for understanding senses come from the continuous transform and the sampling of continuous signals, so these are the foci of the present chapter. The Fourier Transform (FT) then, is the method of finding the frequency components of a signal. There is a variety of different ways to define it, and we’re going to have to leave the mathematical niceties to the many excellent books specialising in the area, such as Gaskill (1978) and Bracewell (1999). We will use what is sometimes called the real form. It’s actually not the mathematically most elegant, but it has the advantage of relating directly to the equations for a real wave form, for a real function, f (x ) 7 . 7 x will often be either time for an auditory signal, or space for an image; we refer to the domain of f (x ) as the time or space domains and the transform as the frequency domain.
eBook - PDF- Ariel Lipson, Stephen G. Lipson, Henry Lipson(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
4 Fourier theory J. B. J. Fourier (1768–1830), applied mathematician and Egyptologist, was one of the great French scientists working at the time of Napoleon. Today, he is best remembered for the Fourier series method, which he invented for representation of any periodic function as a sum of discrete sinusoidal harmonics of its fundamen-tal frequency. By extrapolation, his name is also attached to Fourier transforms or Fourier integrals , which allow almost any function to be represented in terms of an integral of sinusoidal functions over a continuous range of frequencies. Fourier methods have applications in almost every field of science and engineering. Since optics deals with wave phenomena, the use of Fourier series and transforms to analyze them has been particularly fruitful. For this reason, we shall devote this chapter to a discussion of the major points of Fourier theory, hoping to make the main ideas sufficiently clear in order to provide a ‘language’ in which many of the phenomena in the rest of the book can easily be discussed. More complete discussions, with greater mathematical rigour, can be found in many texts such as Brigham (1988), Walker (1988) and Prestini (2004).
eBook - PDFApplied Structural and Mechanical Vibrations
Theory and Methods, Second Edition
- Paolo L. Gatti(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
21 Chapter 2 Mathematical preliminaries 2.1 INTRODUCTION The purpose of this chapter is twofold: (1) to introduce and discuss in some detail a number of mathematical aspects that will be used in the following chapters, and (2) to provide the reader with some concepts, notions and results that are prerequisites for a more advanced and more mathematically oriented approach to the subject matter of this text. In this light, some sec-tions of this chapter can be skipped in a first reading and considered only after having read the subsequent chapters, in particular Chapters 5–9. However, not all the needed results are given here, and some specific topics will be discussed separately in other parts of the book. More spe-cifically, finite-dimensional vector spaces and matrices form the subject of Appendix A, while the fundamental ideas of probability theory – serving as preliminaries to the subject of random vibrations (Chapter 12) – are considered in Chapter 11. Also, when short remarks do not significantly interfere with the main line of reasoning, brief mathematical digressions will be made whenever needed in the course of the text. 2.2 FOURIER SERIES AND FOURIER TRANSFORM In general terms, Fourier analysis is a mathematical technique that deals with two problems: the addition of harmonic oscillations to form a result-ant ( harmonic synthesis ); and the reverse problem of harmonic analysis : given a resultant, find the harmonic oscillations from which it was formed. As a very simple example, it is evident that adding two harmonic oscilla-tions of the same frequency ω 0 f t A t B t f t A t B t 1 1 0 1 0 2 2 0 2 0 ( 29 = + ( 29 = + sin cos , sin cos ω ω ω ω (2.1) results in a harmonic oscillation f ( t ) = f 1 ( t ) + f 2 ( t ) of the same frequency with a sine amplitude A 1 + A 2 and a cosine amplitude B 1 + B 2 . This is not
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