Fourier analysis periodic function

This analysis is based on the fact that any periodic function can be approximated, at any order, by a Fourier series ... 

Fourier analysis makes it possible to analyse a sectionally continuous periodic function into an infinite series of harmonics. For a non-periodic absolutely integrable function, the summation over discrete frequencies becomes an integral... 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

A Fourier synthesis, illustrated in Figure 6.7, is the reverse of a Fourier analysis. A Fourier synthesis involves the summation of waves of known frequency, amplitude and phase in order to obtain a more complicated, but still periodic function. The relationship between a Fourier synthesis and a Fourier analysis is evident from the use of the same set of waves in both Figures 6.6 and 6.7. In a Fourier synthesis, if everything except the relative phases of the component waves are known, there will still be an almost infinite number of ways in which the waves can be combined. 

Fourier analysis The breaking down of a periodic function into its component cosine and sine waves (harmonics) with different amplitudes and frequencies. 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

Optical media that can record elementary gratings, such as photorefractive polymers, are in high demand. In many fields, the decomposition of a signal into a superposition of harmonic functions (Fourier analysis) is a powerful tool. In optics, Fourier analysis often provides an efficient way to implement complex operations and is at the basis of many optical systems. For instance, any arbitrary image can be decomposed into a sum of harmonic functions with different spatial frequencies and complex amplitudes. Each of these periodic functions can be considered... 

A Fourier analysis involves breaking down a periodic (repetitive) function into its component waves, and then deriving the amplitude, frequency, and phase of each of these component waves. In effect, a Fourier analysis takes place in the diffraction experiment when the scattering of X rays by the electron density in the crystal produces Bragg reflections, each with a different amplitude F(hklj and a relative phase a. Thus, X-ray diffraction gives the components to be summed to give an electron density map (without the necessary relative phase angles). 

For more realistic models of the diatom potential, for instance a Morse function, the dynamics is more complicated. For a nonrotating Morse potential, the necessary analytical equations have been worked out (8). For a rotating diatom, however, the problem is more complex. The best approach in this case is one based on the concept of action-angle variables for periodic motions (1). Porter et al. (19) give one recipe. More recently, Eaker (20) has proposed a method based on a Fourier analysis (21) of the motion. 

A Fourier transform enables one to convert the variation of some quantity as a function of time into a function of frequency, and vice versa. Thus, if we represent the quantity that varies in time as x(f), then Fourier analysis enables us to also represent that quantity as a function X i>), where i/ is the frequency (—oo < i/ < oo). Fourier analysis is usually introduced by considering functions that vary in a periodic manner with time which can be written as a superposition of sine and cosine functions (a Fourier series see Section 110.8). If the period of the fvmction x f) is r then the cosine and sine terms in the Fourier series are functions of frequencies 27m/r, where n can take integer values 1, 2, 3, ... 

Harmonic analysis, or Fourier analysis, is the decomposition of a periodic function into a sum of simple periodic components. In particular, Fourier series are expansions of periodic functions /(x) in terms of an infinite sum of sines and cosines of the form... 

Fourier analysis The representation of a function f(x), which is periodic in as an infinite series of sine and cosine functions,... 

The correlation will be maximal if one signal can be displaced with respect to the other until they fluctuate together. The correlation function c(t) will be a more or less noisy sine wave symmetrical around t = 0. The decay of the amplitude envelope from t = 0 indicates the degree of correlation the slower the decay, the higher the correlation. If fi(t) = f2(t), autocorrelation is done by delaying a copy of the function itself and perform the integration of Eq. 8.34. The process will be much the same as a Fourier analysis, a search for periodicity. 

The lowest frequency occurs when n = I and is called the fundamental. Doubling the frequency corresponds to raising the pitch by an octave. Those solutions having values of n > I are known as the overtones. As mentioned previously, one important property of waves is the concept of superposition. Mathematically, it can be shown that any periodic function that is subject to the same boundary conditions can be represented by some linear combination of the fundamental and its overtone frequencies, as shown in Figure 3.8. In fact, this type of mathematical analysis is known as a Fourier series. Thus, while the note middle-A on a clarinet, violin, and piano all have the same fundamental frequency of 440 Hz, the sound (or timbre) that the different instruments produce will be distinct, as shown in Figure 3.9. 

The advantage of the EFT is that this analysis allows one to determine the response of each periodic function when their sum is applied. It should be stressed, however, that the frequency information is for/between/mm = 1/T and the Nyquist frequency/max = l/2At. For example, the FT of the curve displayed in Fig. 2.15 shows that it is composed of four cosine (only real values in the Fourier domain) and three sine functions (only imaginary values in the Fourier domain) (Fig. 2.16). 

Fourier Analysis The Fourier Series for Continuous-Time Periodic Functions The Fourier Transform for Continuous-Time Aperiodic Functions The Fourier Series for Discrete-Time Periodic Functions The Fourier Transform for Discrete-Time Aperiodic Functions Example Apphcations of Fourier Waveform Techniques... 

The occurrence of unstable current conditions is connected with dynamical development features of initial slow disturbances of the velocity profile. The stability analysis is aimed at determining the conditions which promote nonlimited disturbance rising in connection with space and time coordinates. The theoretically based [7,8] proposal is applied to systems with induced or spontaneous convection where two-dimensional disturbances are responsible for stabiUty disruption (such disturbances have a maximum velocity at the initial stage at least). These two-dimensional disturbances can be interpreted as periodic functions of coordinates and then it is possible to decompose the given disturbance into a Fourier series. For example, in the case of an initial spontaneous convective current along the flat vertical plate y=0 the disturbed current function y) x,y,t) is written as [9] ... 

Not all bioelectric potentials are periodic. In many cases, single pulse signals are observed, such as neural response to a single stimulus. The Fourier-series method of spectral analysis does not apply in such cases as it is defined for periodic functions only. It can be extended, however, through a series of limiting processes to develop the Fourier transform integrals which are defined below. 

Fourier Expansions for Basic Periodic Functions The Fourier Transforms Series Expansion Vector Analysis... 

Additive synthesis is deeply rooted in the theory of Fourier analysis. The technique assumes that any periodic waveform can be modelled as a sum of sinusoids at various amplitude envelopes and time-varying frequencies. An additive synthesiser hence functions by... 

This expression covers two methods, Fourier analysis and Fourier synthesis. In almost all cases, it is used to mean Fourier analysis, whereby a periodic function is transformed into the sum of a series of sine waves with different frequencies. The result of the transformation consists of intensities and phases of sine waves. Fourier analysis is widely used in nuclear magnetic resonance and infrared spectroscopy to evaluate the intensities and frequencies of the absorbed waves while the sample is irradiated with equal intensities of all frequencies in a specified range. In such applications, the fast Fourier transformation (FFT) algorithm is most commonly used. The frequencies of the sine waves are harmonic and have a base frequency whose period is the measured time of the periodic function. Use of the FFT requires the values of the periodic function at several times in equal intervals. The number of data points has to be a power of two. 

Fourier analysis treats the representation of periodic functions as hnear combinations of sine and cosine basis functions. In chemical engineering, Fourier analysis is applied to study time-dependent signals in spectroscopy and to analyze the spatial structure of materials from scattering experiments. Here, the basic foundation of Fourier analysis is presented, with an emphasis upon implementation in MATLAB. 

We begin our discussion of Fourier analysis by considering the representation of a periodic function f t) with a period of 2P, f t + 2P) = f t). If f(f) has a finite number of local extrema and a finite number of times tj e [0,2P] at which it is discontinuous, Dirichlet s theorem states that it may be represented as the Fourier series... 

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