Frequency function

In practical applications, x(t) is not a continuous function, and the data to be transformed are usually discrete values obtained by sampling at intervals. Under such circumstances, I hi discrete Fourier transform (DFT) is used to obtain the frequency function. Let us. suppose that the time-dependent data values are obtained by sampling at regular intervals separated by [Pg.43]

The cumulative-frequency function calculated from this simple expression is compared with the precise value in Table 9-10. 

For this model the frequency function of the bomber s optimal strategy, as obtained through game theoretic analysis, is... 

Fig. 4-4 The age frequency function j/ x) and the residence time frequency function 4> x) and the corresponding average values and r, for the three cases described in the text (a) ta > t, (b) la = xy, (c) ta > t,. (Adapted from Bolin and Rodhe (1973) with permission from the Swedish Geophysical Society.)...

It is normally called the differential distribution function (of residence times). It is also known as the density function or frequency function. It is the analog for a continuous variable (e.g., residence time i) of the probabiUty distribution for a discrete variable (e.g., chain length /). The fraction that appears in Equations (15.2), (15.3), and (15.6) can be interpreted as a probability, but now it is the probability that t will fall within a specified range rather than the probability that t will have some specific value. Compare Equations (13.8) and (15.5). 

Fourier transform of the time function x(t) into the frequency function X(a>) ... 

Randomness can be made objective by measuring some random variables over it and then performing an appropriate statistics. Let us consider the interdistances between subsequent time positions (Ax) in a superimposed pattern they are clearly not regular and can be employed to characterize randomness (see Fig. 4.2a). In fact, when their frequency function is computed (Fig. 4.2b) a distinct and familiar regularity singles out the exponential function. 

The probability density function f(x) (also the density function or frequency function) is the... 

E(t) is a probability density function or frequency function and E(f)df is the fraction of material which leaves the system with an age of between t and (t + df) units of time. Since all material must have a residence time between zero and infinity... 

Cumulative distribution function (CDF) The CDF is referred to as the distribution fnnction, cumulative frequency function, or the cnmnlative probability fnnction. The cumnlative distribution fnnction, F(x), expresses the probability that a random variable X assumes a value less than or eqnal to some valne x, F(x) = Prob (X > x). For continnons random variables, the cnmnlative distribution function is obtained from the probability density fnnction by integration, or by snmmation in the case of discrete random variables. 

Probability density function (PDF) The PDF is referred to as the probability function or the frequency function. For continuous random variables, that is, the random variables that can assume any value within some defined range (either finite or infinite), the probability density function expresses the probability that the random variable falls within some very small interval. For... 

Figure 4.9 Comparison between SIMS measured profiles (dotted line) and computed profiles using the Pearson frequency functions with the first four moments determined for each individual profile (solid line) or computed via analytical functions obtained through the best fit of the individual moments (long dashed line). (From [70]. 2003 American Institute of Physics. Reprinted with permission.)...

FIG. C.1 A normal, or Gaussian, distribution (a) represented as a frequency function (b) represented as a cumulative function and (c) represented as a cumulative function linearized by plotting on probability paper. 

Since NMR spectra are not sequences of lines representing discrete Larrnor frequencies but sequences of Lorentzian frequency distributions f(to) (Fig. 1.9), eq. (2.10) must be replaced by eq. (2.11) M0 sin c is multiplied by the frequency function f(to), where a> represents the difference between the frequency ojx and the Larrnor frequency distribution con + Aw, w = co1 — (w0 + Aro). Further, Mosin0f(tu)e must be integrated over the Larrnor frequency distribution. Given a Lorentzian line shape as in Fig. 1.9, the limits of integration are oo ... 

The frequency function is a Lorentzian with linewidth at half height of (jt72)-i(= Ri/n). The same, of course, holds in the continuous wave experiment. R2 is a measure of the uncertainty of the energy levels, which gives the linewidth in every spectroscopy. The uncertainty principle, according to which the uncertainty in energy of a level is inversely proportional to the lifetime, tells us that T2 is a measure of the lifetime of the energy levels. 

The measured droplet size distribution is represented in terms of the number frequency function of droplets, which is defined as... 

In order to obtain the frequency function of the amplitudes for a fixed value W one now has to integrate over W the product of the secular behaviour of jj and the Porter-Thomas distribution. The distribution function of the amplitudes for arbitrary values of initial and final energies then follows after a second integration [VER79]... 

The sign in the present instance is negative if we regard q as the number of particles greater than a given diameter d. We are not concerned with the form of either q or F(d) for the time being. The technique involved requires only that we determine the size-distribution graphically. With sedimentation methods the form of frequency function is important mathematically only in so far as it explains the relationships of the variables measured. 

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

---