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By Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

ISBN-10: 1483197603

ISBN-13: 9781483197609

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27) -cp(t) = x(t). 29) Let us p u t Then t h e function which has zero mathematical expectation, will possess t h e same correlation function as X(t). Accordingly, in t h e deriva­ tion of the general properties of correlation functions it is possible without loss of generality to consider the mathemat­ ical expectation of the random function as equal to zero. We will make extensive use of this fact later in order to shorten proofs. § 6. T H E DIFFERENTIATION AND INTEGRATION OF RANDOM FUNCTIONS We introduce t h e concept of t h e derivative of a random function.

In this case the function K(t) giving the value of the covariance between X(t2) and X(^) will tend to zero as | r | — oo. Hence K(t) may either decrease monotonically as represented in Fig. 4 or be of the form represented in Fig. 5. K(T) FIG. 4. 30 RANDOM FUNCTIONS K(x) FIG. 5. Functions of the form shown in Fig. 21) and a function of the form shown in Fig. 5 by the expression K(r) ^ o2e~« ITI cos £T or by the expression K(r) ^ a2e-^z cos &r. 23) In what follows we shall find it necessary to approximate the correlation function by a sufficiently simple analytic ex­ pression when solving a number of problems; the accuracy of the approximation required will be discussed.

K(D _ "" -cr'oc FIG,. 6. "" dK(T) dx 37 GENERAL PROPERTIES FIG. 7. On t h e other hand, the first derivatives of the functions a2e~*lxi and a2e~*lz* cos /^r remain continuous for any values of r and accordingly, correlation functions of this t y p e correspond to differentiable random processes. As an additional example of a correlation of a differen­ tiable process consider the expression y (6. 13) o2e-« I TI sin /? 14) a2e~a ITI ( cos j5r+-5-sin/S| r\ K(r) the first derivative of which dK(x) dx a 2 + /S2 P is continuous a t zero and consequently the second derivative exists a t this point.

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