Function characteristic

Sensitive parameters are necessary to compare several high resolution magnetic field sensors. Such parameters can be found with methods of signal theory for LTI-systems. The following chapter explains characteristic functions and operations of the signal analysis for linear local invariant systems and their use in non-destructive testing. 

Figure 1 Characteristic functions of LLI-systems and graphical superposition...

Methods from the theory of LTI-systems are practicable for eddy-current material testing problems. The special role of the impulse response as a characteristic function of the system sensor-material is presented in the theory and for several examples. 

Substitutive Nomenclature. The first step is to determine the kind of characteristic (functional) group for use as the principal group of the parent compound. A characteristic group is a recognized combination of atoms that confers characteristic chemical properties on the molecule in which it occurs. Carbon-to-carbon unsaturation and heteroatoms in rings are considered nonfunctional for nomenclature purposes. 

The polyimides have the characteristic functional group below and are thus closely related to the polyamides. However, the branched nature of the... 

Eigen-farbstoff, m. self-dye, self-color, -form, /. characteristic form, true form, -frequenz, /. proper, characteristic, or natural frequency, -funktion, /. proper function, characteristic function, eigenfunktion. -gewicht, n. 

If we compare different liquids at temperature T, there will be a characteristic function p(r) for each liquid. With rise of temperature, as the Brownian motion becomes more lively, p(r) will doubtless change in yidu... 

The Characteristic Function.—The calculation of moments is often quite tedious because of difficulties that may be encountered in evaluating the pertinent integrals or sums. This problem can be simplified quite often by calculation of the so-called characteristic function of the distribution from which, as we shall see, all moments can be derived by means of differentiation. This relationship between the characteristic function and moments is sufficient reason for studying it at this time however, the real significance of the characteristic function will not become apparent until we discuss the central limit theorem in a later section. 

The characteristic function of the random variable (or, equivalently, of the distribution function F ) is defined to be the expectation of the random variable eiv4> where v is a real number.16 In symbols... 

The characteristic function is thus seen to be the Fourier transform17 of the probability density function p+. The fact that the function etv< is bounded, e<0< = 1, implies that the characteristic function of a distribution function always exists and, moreover, that... 

One of the most important properties of Fourier transforms and, consequently, of characteristic functions, is their invertibility. Given a characteristic function M, one can calculate the probability density function p by means of the inversion formula... 

Thus, knowledge of the characteristic function of a distribution, unlike... 

The technique for calculating the moments of a random variable from the characteristic function of the random variable can be derived by first differentiating both sides of Eq. (3-77) n-times with respect to v... 

As an example of these techniques, we shall calculate the characteristic function of the gaussian distribution with zero mean and unit variance and then use it to calculate moments. Starting from the definition of the characteristic function, we obtain18 ... 

This result checks with our earlier calculation of the moments of the gaussian distribution, Eq. (3-66). The characteristic function of a gaussian random variable having an arbitrary mean and variance can be calculated either directly or else by means of the method outlined in the next paragraph. 

The calculation of characteristic functions is sometimes facilitated by first normalizing the random variable involved to have zero mean and unit variance. The transformation that accomplishes this is... 

Equation (3-88) enables us to calculate the characteristic function of the unnormalized random variable from a knowledge of the characteristic function of . For example, the characteristic function of a gaussian random variable having arbitrary mean and variance can be written down immediately by combining Eqs. (3-83) and (3-88)... 

We conclude this section with a derivation of the characteristic function of the Poisson distribution. Starting from the definition, Eq. (3-77), we obtain... 

A table giving the characteristic function, mean, and variance of some important probability distributions appears in Fig. 3-6. 

Distribution Probability Density Function Characteristic Function Mean Variance... 

Fig. 3-6. Some Typical Characteristic Functions, Means, and Variances.

Joint Moments and Characteristic Functions.—The joint momenta cckli...ikn of a family of n random variables dm are defined by the expression... 

The joint characteristic function is thus seen to be the -dimensional Fourier transform of the joint probability density function The -dimensional Fourier transform, like its one-dimensional counterpart, can be inverted by means of the formula... 

In other words, knowledge of the joint characteristic function of a family of random variables is tantamount to knowledge of their joint probability density function and vice versa. 

The joint characteristic function is related to the characteristic functions of the individual random variables by means of the formula... 

One of the uses of the joint characteristic function is in the calculation of joint moments. In order to simplify the notation, we shall only... 

Further, and considerably more important, applications of joint characteristic functions will be given in later sections. 

Another important result states that the characteristic function of a sum of statistically independent random variables is the product of the characteristic functions of the individual summands. The reader should compare this statement with the deceptively similar sounding one made on page 154, and carefully note the difference between the two. The proof of this statement is a simple calculation... 

The proof of the central limit theorem begins with the calculation of the characteristic function of... 

Now, since the random variable — m /jj has finite mean (=0) and variance (= 1), both its characteristic function and the logarithm of its characteristic function have finite first and second derivatives. It follows that In Mi1 mi)/ffl(i ) can be expanded in a Taylor series with remainder42 as follows43... 

The right-hand side of Eq. (3-205) is the characteristic function of the gaussian distribution having zero mean and unit variance, and this leads us to conclude that the distribution function of sj approaches... 

The multidimensional central limit theorem now states that the multidimensional characteristic function of sfn, -sj, behaves as follows ... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

Although we cannot easily obtain expressions for the probability density functions of Y(t), it is a simple matter to calculate its various moments. We shall illustrate this technique by calculating all possible first and second moments of Y(t) i.e., E[7(t)] and E[Y(t)Y(t + r)], — oo < v < oo. The pertinent characteristic function for this task is MYQt (hereafter abbreviated MYt) given by... 

The Gaussian Process.—A gaussian process was defined in the last section to be a process all of whose finite-order distributions are multi-dimensional gaussian distributions. This means that the multi-dimensional characteristic function of Px.fK must be of the form... 

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