Fourier cosine function

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

Fourier coefficients, represent the frequencies of the cosine functions. In the remainder of this chapter we use the shorthand notation F(n) for [A - jB ]. 

The term Fourier coefficient originates from the theory of Fourier series, in which periodic functions are expanded based on a set of sine- and cosine-functions. The expansion coefficients are called Fourier coefficients. 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

Conversely, a rf field is totally correlated because it is represented by a sine (or cosine) function and, as a consequence, its value at any time t can be predicted from its value at time zero. The efficiency of a random field at a given frequency co can be appreciated by the Fourier transform of the above correlation function... 

Fig. 13 Simulation of a monochromatic source with a finite arm displacement of the interferometer, (a) Truncated cosine function (30 discrete data points), (b) Its Fourier transform, the sine function, which simulates the infrared spectral line.

Assuming that the rate of change of the optical retardation introduced by the interferometer is the same for all the input radiation frequencies (which is normally the case), each individual value of v in the broadband source output contributes a different value of /m in the a.c. component of the detector output (see Figure 3.20). These contributions are, of course, summed by the detector. The combined detector output (see Figure 3.21) arising from the simultaneous measurement of all modulated input signals, is a sum of cosine functions. Such a sum is a Fourier series, and the amplitudes of the individual... 

In Fourier compression, each profile (x ) is essentially decomposed into a linear combination of sine and cosine functions of different frequency. If the spectrum x is considered to be a continuous function of the variable number m, then this decomposition can be expressed as ... 

The basis set using a finite basis set necessarily leads to an inexact wavefunc-tion, in much the same way that representing a function by a finite Fourier series of sine and cosine functions necessarily gives an approximation (albeit perhaps an excellent one) to the function. 

Let us note for future purposes that since Cvv(o)) is an even function of to, Eq. (191) can be rewritten as a Fourier cosine integral,... 

This is a Fourier series cxp2uision that exprfcsses a constant in terms of an infinite series of cosine functions. Now we multiply both sides of Eq. 4—21 by cos(A X), and integrate fromX = 0 to X = 1. The right-band side involves an infinite number of integrals of the form /Jcos(A X) cos(A X)< v, It can be shown that all of these integrals vanish except when n = r i. and the coefficient v4, becomes... 

In this Figure we have chosen a cenirosymmetric function, g y), that is a cosine function with a periodicity of 1. The integration has been approximated by a summation with small increments in y in order to demonstrate how a Fourier transform can be calculated. 

In other words, the Chebyshev polynomials are essentially a cosine function in disguise. This duality underscores the effectiveness of the Chebyshev polynomials in numerical analysis, which has been recognized long ago by many,[7] including the great Hungarian applied mathematician C. Lanczos.[9] In particular, Fourier transform (and FFT) can be readily implemented in the spectral method involving the Chebyshev polynomials. 

One solution to the problem is to increase the ionization probability. This can be done by choosing primary ions with heavy mass, for example, Bi+ or even Ccarbon atoms. The noise level can also be reduced by techniques of digital image processing. For example, a fast Fourier transform technique has been used to remove noise from the image. This technique transforms an image from a space domain to a reciprocal domain by sine and cosine functions. Noise can be readily filtered out in such domain. After a reverse Fourier transform, filtered data produces an image with much less noise. 

In the transformation the physical units are inverted. When the interferogram is expressed in optical path difference units (cm), the spectrum is obtained in wave-numbers (cm-1) and when the interferogram is expressed in time units (s) the spectrum is in frequency units (s 1). Apart from sine and cosine functions, box-car and triangular, etc. functions are also known, for which the Fourier transformation can be calculated. When applying the Fourier transformation over the whole area + oo, the arm of the interferometer also would have to be moved from — co to +co. When making a displacement over a distance of +L only, the interferogram has to be multiplied by a block function, which has the value of 1 between + and —I and the value 0 outside. I then influences the resolution that can be obtained. 

Each diffraction spot is caused by reflection of X-rays by a particular set of planes in the crystal. If the crystal contains layers of atoms with the same spacing and orientation as a particular set of planes which would satisfy Bragg s law (if the set of planes is physically present), the corresponding diffraction spot will be strong. On the other hand, if only few atoms in a crystal correspond to a particular set of planes, the corresponding reflection will be weak. The complicated structure present in the crystal is transformed by the diffraction process into a set of diffraction spots which correspond to sets of planes (more precisely, sinusoidal density waves), just as our ear converts a complicated sound signal into a series of (sinusoidal) tones when we listen to music. This conversion of a complicated function into a series of simple sine and cosine functions is called a Fourier transformation. 

By Fourier transformation, a signal is decomposed into its sine and cosine components [Angl]. In this way, it is analysed in terms of the amplitude and the phase of harmonic waves. Sine and cosine functions are conveniently combined to form a complex exponential, coscot 4- i sinwt = exp icomplex amplitudes of these exponentials constitute the spectrum F((o) of the signal f(t), where co = In IT is the frequency in units of 2n of an oscillation with time period T. The Fourier transformation and its inverse are defined as... 

The Fourier transform of a decaying cosine function cosQt exp(-f/7 2) is an absorption mode Lorentzian centred at frequency 2 the real part of the spectrum has been plotted. 

The Ruben/State/Haberkorn mode [2.40] This quadrature detection mode leads to phase sensitive 2D spectra and is based on the different symmetry properties of the sine and cosine functions after Fourier transformation. As shown in equation [2-15] the first term of imaginary part of the sine data set Im[s(coi, CO2)] has a negative sign in contrast to the real part of the cosine data set. Simple mathematical addition of the real part and the imaginary part of the sine and cosine modulated data sets after Fourier transformation... 

A Fourier series is an infinite series of terms that consist of coefficients times sine and cosine functions. It can represent almost any periodic function. 

If we want to produce a series that will converge rapidly, so that we can approximate it fairly well with a partial sum containing only a few terms, it is good to choose basis functions that have as much as possible in common with the function to be represented. The basis functions in Fourier series are sine and cosine functions, which are periodic functions. Fourier series are used to represent periodic functions. A Fourier series that represents a periodic function of period 2L is... 

Fourier series occur in various physical theories involving waves, because waves often behave sinusoidally. For example, Fourier series can represent the constructive and destructive interference of standing waves in a vibrating string. This fact provides a useful way of thinking about Fourier series. A periodic function of arbitrary shape is represented by adding up sine and cosine functions with... 

If the function f x) is an even function, all of the bn coefficients will vanish, and only the cosine terms will appear in the series. Such a series is called a Fourier cosine series. If f x) is an odd function, only the sine terms will appear, and the series is called a Fourier sine series. If we want to represent a function only in the interval 0 < a < L we can regard it as the right half of an odd function or the right half of an even function, and can therefore represent it either with a sine series or a cosine series. These two series would have the same value in the interval 0 < X < L but would be the negatives of each other in the interval — L < jc < 0. 

In the previous example the transform integral was separated into one part containing a cosine function and one containing a sine function. If the function f x) is an even function, its Fourier transform is a Fourier cosine transform ... 

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