The power spectrum

Consider a general stationaiy stochastic process x(Z), a sample ofwhich is observed in the interval 0 t T. Expand it in Fourier series 

If x(Z) is real then = x . Equation (7.69) resolves x(Z) into its spectral components, and associates with it a set of coefficients x such that x p is the strength or intensity of the spectral component of frequency However, since each realization of x(Z) in the interval 0. T yields a different set x , the variables x are themselves random, and characterized by some (joint) probability function P( x ). This distribution in turn is characterized by its moments, and these can be related to properties of the stochastic process x(Z). For example, the averages x satisfy 

Note that from Eq. (7.70) xo = (l/T) /q x t) = x is the time average ofx(Z) for any particular sampling on the interval T. Foran ergodic process limr- co. = x) we thus find that Xo xo = x.  

For our purpose the important moments are ( x p), sometimes referred to as the average strengths of the Fourier components The power spectrum I (co) of the stochastic process is defined as the T - oo limit of the average intensity at frequency m  

Note that, as defined, I co is a real function that satisfies I —co) = I co). 

Problem 7.10. Show that another expression for the power spectrum is 

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In the linear approximation there is a direct Fourier relationship between the FID and the spectrum and, in the great majority of experunents, the spectrum is produced by Fourier transfonnation of the FID. It is a tacit assumption that everything behaves in a linear fashion with, for example, imifonn excitation (or effective RF field) across the spectrum. For many cases this situation is closely approximated but distortions may occur for some of the broad lines that may be encountered in solids. The power spectrum P(v) of a pulse applied at Vq is given by a smc fiinction 18]... 

It remains to be seen, if the approximation using large time steps is reasonable. We shall show later the effect of the approximation on the power spectrum of the trajectory. More specifically, we shall demonstrate that large time steps filter out high frequency motions. 

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. 

At the other end of the power spectrum, there is increasing interest in fuel cells for small electronic... 

Step 5 - Calculate the Power Spectrum P f) Having calculated R t), we... 

Using equation 6.16 we obtain the (continuous form of the) power spectrum ... 

Notice that P f) has a maximum at f — 1/2. The presence of this maximum can be directly traced back to the length-2 loop of the STG shown above, which yields a period-2 component in the spatial configurations. The half-width of the peak (=log(l/g)) is seen to decrease as q decreases, a natural consequence of the period-2 part of the STG being visited more often as q increases. Figure 6.4 shows in empirical estimate of the power spectrum for rule R56 obtained for a size iV = 2 = 2048 system. 

It is not hard to see that, given a power-law distribution of lifetimes (equation 8.113), the power spectrum S f), defined by... 

Assuming an isotropic Gaussian distribution with normalization, we have the actual form of the power spectrum,... 

It is often useful to deal with the statistics in Fourier space. The Fourier transform of the correlation is called the power spectrum... 

For Kolmogorov statistics, it turns out that the power spectrum is infinite at the origin, which means that the variance is infinite. The structure function can be used instead of the co-variance to overcome this problem. It is defined as... 

This is the autocorrelation and by the Wiener-Khintchine theorem the power spectrum of the disturbance is given by its Fourier transform,... 

Fig. 6 shows the FFT spectrum for calculated bed pressure drop fluctuations at various centrifugal accelerations. The excess gas velocity, defined by (Uo-U ,, was set at 0.5 m/s. Here, 1 G means numerical result of particle fluidization behavior in a conventional fluidized bed. In Fig. 6, the power spectrum density function has typical peak in each centrifugal acceleration. However, as centrifugal acceleration increased, typical peak shifted to high frequency region. Therefore, it is considered that periods of bubble generation and eruption are shorter, and bubble velocity is faster at hi er centrifugal acceleration. 

The phase spectrum 0(n) is defined as 0(n) = arctan(A(n)/B(n)). One can prove that for a symmetrical peak the ratio of the real and imaginary coefficients is constant, which means that all cosine and sine functions are in phase. It is important to note that the Fourier coefficients A(n) and B(n) can be regenerated from the power spectrum P(n) using the phase information. Phase information can be applied to distinguish frequencies corresponding to the signal and noise, because the phases of the noise frequencies randomly oscillate. 

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//... 

Fig. 40.18. Noise characterisation in the frequency domain. The power spectrum IF(v)l of three types of noise, (a) White noise, (b) Flicker or 1//noise, (c) Interference noise.

If a small amount of gramicidin A is dissolved in a BLM (this substance is completely insoluble in water) and the conductivity of the membrane is measured by a sensitive, fast instrument, the dependence depicted in Fig. 6.15 is obtained. The conductivity exhibits step-like fluctuations, with a roughly identical height of individual steps. Each step apparently corresponds to one channel in the BLM, open for only a short time interval (the opening and closing mechanism is not known) and permits transport of many ions across the membrane under the influence of the electric field in the case of the experiment shown in Fig. 6.15 it is about 107 Na+ per second at 0.1 V imposed on the BLM. Analysis of the power spectrum of these... 

The nerve axon sodium channel was studied in detail (in fact, as shown by the power spectrum analysis, there are two sorts of this channel one with fast opening and slow inactivation and the other with opposite properties). It is a glycoprotein consisting of three subunits (Fig. 6.22), the largest (mol. wt. 3.5 X 105) with a pore inside and two smaller ones (mol.wt 3.5 X 104 and 3.3 X 104). The attenuation in the orifice of the pore is a kind of a filter... 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

Several algorithms have been used to determine surface fractal dimension from SPM images. The most popular ones are the power-spectrum method,1 2 68"71 the triangulation method,10 37 ... 

Because simulated water is a classical liquid, the computed power spectrum which describes the translational motions, is bound to disagree with that of real water. Figure 37, shows that the power spectrum has peaks at 44 cm-1 and 215 cm-1, whereas for real water they occur at 60 cm-1 and 170 cm-1. A similar discrepancy exists between simulated and real water rotational power spectra (compare the simulated water frequencies 410 cm-1, 450 cm-1 and 800-925 cm-1 with the accepted experimental values 439 cm-1, 538 cm-1 and 717 cm-1). In this model localization of the molecules around their momentary orientations is only marginal. 

From the study presented in this chapter, it has been demonstrated that a CSTR in which an exothermic first order irreversible reaction takes place, can work with steady-state, self-oscillating or chaotic dynamic. By using dimensionless variables, and taking into account an external periodic disturbance in the inlet stream temperature and coolant flow rate, it has been shown that chaotic dynamic may appear. This behavior has been analyzed from the Lyapunov exponents and the power spectrum. 

A spectrum is the distribution of physical characteristics in a system. In this sense, the Power Spectrum Density (PSD) provides information about fundamental frequencies (and their harmonics) in dynamical systems with oscillatory behavior. PSD can be used to study periodic-quasiperiodic-chaotic routes [27]. The filtered temperature measurements y t) were obtained as discrete-time functions, then PSD s were computed from Fast Fourier Transform (FFT) in order to compute the fundamental frequencies. 

The power spectrum density (PSD) is a widely used tool to find fundamental frequencies (and their harmonics) in d mamical systems with oscillatory... 

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