Spectrum function

Finding the values of G allows the determination of the frequency-domain spectrum. The power-spectrum function, which may be closely approximated by a constant times the square of G f), is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. From the power spectra, broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensemble averaging, which is a point-by-point average of a squared-spectra set. 

The absorption edge spectrum function expresses p, the TiC-absorption coefficient, in terms of Ex. n is defined here by normalization conditions and the equation ... 

Turbulent flow structure energy spectrum function, scale up... 

Ogawa, K. (1988). Energy spectrum function of eddy group gathered together as a model of turbulence. Ini. J. Eng. Fluid Mech., 1, 235-244. 

Ogawa, K. (1992). Evaluation of common scaling-up rules for a stirred vessel from the viewpoint of energy spectrum function. J. Chem. Eng. Jpn, 25, 750-752. 

The real power of the Fourier transform is the linear nature of the calculation. If we have an FID that is a sum of two different pure frequencies (like Fig. 3.16), the spectrum function looks like this ... 

Whereas it seems justified to assume that the most unsaturated parts of the surface of iron catalysts are the most active in the synthesis, it may well be that on Mo and W, which are supposed to bind nitrogen stronger, an intermediate part of the energy spectrum functions most actively in the catalytic reaction, and it may therefore be more nearly correct to say that in this case surface nitrides are the catalysts. 

Dimethoxycanthin-6-one has been isolated from Picmsma ailan-thoides Sieb et Zucc. (Simarubaceae), and its structure was established by recognition of the characteristic UV-spectrum, functional group analysis, and oxidation to /J-carboline-1 -carboxylic acid (10). 

Kolmogoroff (KIO) developed the following expression for the energy spectrum function E k) applicable to the inertial subrange ... 

Three-dimensional energy spectrum function Shear energy... 

Further contributions to the subject were made by Taylor in 1938. Two important consequences of the non-linearity of the Navier-Stokes equations were identified First, the skewness of the probability distribution of the difference between the velocities at two points, and the existence of an interaction or modulation between components of turbulence having different length scales. Secondly, the Fourier transform of the correlation between two velocities is an energy spectrum function in the sense that it describes the distribution of kinetic energy over the various Fourier wave-number components of the turbulence [164]. Taylor expressed in mathematical form the relation between the correlation function and the ID spectrum function. 

Hinze ([66], sect 3-4) derived an analytical expression for the 3D spectrum E k,t) as a function of wave number k and time t. This result shows that E k = 0) 0, then the spectrum function increases rapidly, reaches a maximum value, and decreases monotonously to zero as k increases. 

If a homodyne method is used, the measured autocorrelation function (g,x) can be interpreted by using the Siegert relation. Equation 5.454. The translational and rotational diffusion coefficients for several specific shapes of the particles are given in Table 5.9. The respective power spectrum functions can be calculated by using the Fourier transform. Equation 5.449b. 

This vector was the left domain of the spectrum function Gex. The right domain of Gex was an array of frequencies, W. W was usually composed of 50 numbers to obtain a reasonable representation of the spectrum across the width of the terminals typewriter. However, W could be of any useful length subject only to time limitations and the memory size of the workspaces which was 50 kbytes for these calculations. 

After considerable work, it was determined that the hosts which work best in this application are those in which vlbronic coupling is minimized. This was established by determining and comparing the phonon dispersion branches of the various compounds. It was found that those hosts which have low energy optical branches in the phonon spectrum function best for up-conversion phosphor applications. Note that this is akin to minimizing ground state perturbation at the activator site proper choice of host. The best hosts were found to be ... 

A Fourier transform of the autocorrelation function of a stochastic process gives the power spectrum function which shows the strength or energy of the process as a function of frequency [17]. Frequency analysis of a stochastic process is based on the assumption that it contains features changing at different frequencies, and thus it can be described using sine and cosine functions having the same frequencies [16]. The power spectrum is defined in terms of the covariance function of the process, Vk = Cov(e,. et k). as... 

A Fabry-Perot (FP) interferometer can be used in an optical sensing system to pass a specific part of the spectrum, functioning like a monochromator. An FP structure is the most suitable for passband optical filters. A Fabry-Perot optical filter consists of two parallel mirrors with a resonance cavity in the middle (Fig. 2). The equation 7u = 2nd shows its operation principle, where n is the refractive index of the cavity medium, d the cavity length, X the incident wavelength, and q the interference order q = 1,2,3,...). 

Tobolsky and his coworkers found that the H t) relaxation spectrum function for polyisobutylene may be represented by the combination of a wedge and box, i.e., as... 

Denote the generalized bath interaction spectrum functions as... 

It is well known that Caai ) > 0 for all values of u. In Appendix A, we further show that the positivity of spectrum functions can be generalized... 

If we shift the signal shown in Figure 9.1 by a quarter cycle as shown in Figure 9.2, we find a different spectrum function. Again Qq = 0, but the new signal is an even function with quarter-wave symmetry. Thus... 

Figure 9.1. Square wave and associated spectrum functions.

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