Frequency spectrum, approximations

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. 

Ideally, any procedure for signal enhancement should be preceded by a characterization of the noise and the deterministic part of the signal. Spectrum (a) in Fig. 40.18 is the power spectrum of white noise which contains all frequencies with approximately the same power. Examples of white noise are shot noise in photomultiplier tubes and thermal noise occurring in resistors. In spectrum (b), the power (and thus the magnitude of the Fourier coefficients) is inversely proportional to the frequency (amplitude 1/v). This type of noise is often called 1//... 

There are many specific ways to generate equally spaced tags but they are all based on the same principle of manipulating the rf pulses to generate equally spaced bands of rf radiation in the frequency domain. It is well known that under ordinary conditions, meaning normal levels of nuclear spin excitation, the frequency spectrum of the rf excitation pulse(s) is approximately the Fourier transform of the pulses in the time domain. Thus, a single slice can be generated in the... 

In the IR spectrum of nitronate (76b), the X st,etch.c=o band is shifted to lower frequencies by approximately 150 cm-1 accompanied by a decrease in the intensity of this signal by more than twice due to coordination of the boron atom at the carbonyl group. 

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

This treatment is oversimplified because, in addition to neglecting inner sphere contributions to the reorganization energy it approximates the dielectric frequency spectrum to a single frequency, 0jo — 1011 s 1, corresponding to the Debye dielectric relaxation which probably varies in the vicinity of the ions. The cathodic current is given by... 

And we have stated that on account of the atomic nature of the. material the velocity depends on wave length, when the wave length beeomes of atomic dimensions. These limitations mean that the frequency spectrum which wc have so far found is not very close to the truth. Nevertheless, our model is good enough so that valuable results can be obtained from it, and we go on now to describe the approximations made by Debye, loading to the specific heat curve known by his name. 

Debye s Theory and the Parameter y.—Debye s theory furnishes us with an approximation to the frequency spectrum of a solid, and we can use this approximation to find how the frequencies change with... 

They can be used to model the long-time behaviour of a dynamical system. This can be done either by approximating the relevant correlation functions using only the stable portion of the frequency spectrum or by using the unstable lobe of the spectrum for drawing (empirical) relationships to the diffusion constant. " ... 

Absorption bands that are attributed to overtone and combination vibrations are also observed in the IR spectrum of polyatomic molecules. Overtone vibrations occur at frequencies of approximately integral multiples of the fundamental frequencies. Combination vibrations appear at frequencies that are the sum or the difference of the frequencies of two or more frmdamental vibrations. Overtone and combination bands are much less intense than fundamental bands. 

It is a tacit assumption that everything behaves in a linear fashion, for example, that the excitation (or effective rf field) is uniform across the entire spectrum. In many cases this situation is closely approximated but distortions can occur for the broad lines that may be encountered. The frequency spectrum A(v) of a pulse of duration Tp applied at Vo is given by a sine function... 

Successful first-principles molecular dynamics simulations in the Car-Paxrinello framework requires low temperature for the annealed electronic parameters while maintaining approximate energy conservation of the nuclear motion, all without resorting to unduly small time steps. The most desirable situation is a finite gap between the frequency spectrum of the nuclear coordinates, as measured, say, by the velocity-velocity autocorrelation function. 

Cp(T) is the property normally measured by calorimetry. However, CV(T) is also important, since it is directly calculated by theoretical models which express the heat capacity of a material in terms of the vibrational motions of its atoms. Vibrational motions approximated by the harmonic oscillator model are commonly accepted to be the source of the heat capacities of solids. CV(T) is estimated by integrating over the frequency spectrum of these vibrations. 

Obviously, the details in the time-profile, 7, and the frequency spectrum, Fp, of the incident X-pulse, depend on the experimental setup. However, if the duration of the pulse is either sufficiently short or sufficiently long compared to the time scale of the nuclear dynamics, 7 may be replaced by either a delta function or a constant on the nuclear time scale. Likewise, if the width of Fp can be neglected (known as the static approximation ), we can obtain simplified expressions for the differential scattering signal. However, as pointed out earlier, the frequency widths of X-ray pulses obtained from, e.g., synchrotron radiation are typically on the order of percent of the carrier frequency. Hence, in order to simulate the finer details of the experimental signal, the actual frequency distribution of the incident X-ray pulse must be taken into account [29],... 

The combustion roar associated with flares typically peaks at a frequency of approximately 63 Hz while combustion roar associated with burners can vary in the 200-500 Hz range. Burner noise can have a spectrum shape and amplitude that can vary with many factors. Several of these factors include the internal shape of the furnace, the design of the burner muffler, plenum and tile, the acoustic properties of the furnace lining, the transmission of the noise into the fuel supply piping, and the transmissive and reflective characteristics of the furnace walls and stack. 

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