310 BELL SYSTEM TECHNICAL JOURNAL 



proceeding as in section 1.4, will give us the normal distribution plus the first 

 correction term. It is rather a messy affair. An idea of how it looks may 

 be obtained by taking the special case in which F{t) is an even function of / 

 and neglecting terms of type B. Then 



P{a, , ■■■ a,,b,,---hs) = {\+'n)Jl -^—2— (1.7-12) 

 where 



Xn — > 3'" — 



(Tn <Tn 



r, = (IvT)-'" X) [xk+f{xkxe - y,yc) + 2 yk^tykjl] (1.7-13) 

 k.t 



and the summation extends over 2 < k -\- I < N with k < I. 



It is seen that if T and N are held constant, the correction term rj ap- 

 proaches zero as v becomes very large. A very rough idea of the magnitude 

 of t] may be obtained by assuming that unity is a representative value of the 

 x's and j's. Further assuming that there are N terms in the summation 

 each one of which may be positive or negative suggests that magnitude of 

 the sum is of the order of N. Hence we might expect to find that ?j is of 

 the order of N{2vTy^'\ 



PART II 

 POWER SPECTRA AND CORRELATION FUNCTIONS 



2.0 Introduction 



A theory for analyzing functions of time, /, which do not die down and 

 which remain finite as / approaches infinity has gradually been developed 

 over the last sixty years. A few words of its history together with an ex- 

 tensive bibliography are given by N. Wiener in his paper on "Generalized 

 Harmonic Analysis"." G. Gouy, Lord Rayleigh and A. Schuster were led 

 to study this problem in their investigations of such things as white light 

 and noise. Schuster^^ invented the "periodogram" method of analysis which 

 has as its object the discovery of any periodicities hidden in a continuous 

 curve representing meteorological or economic data, 



" Ada Math., Vol. 55, pp. 117-258 (1930). See also "Harmonic Analysis of Irregular 

 Motion," Jour. Math, and Phys. 5 (1926) pp. 99-189. 



'2 The periodogram was first introduced by Schuster in reference 10 cited in Section 

 1.7. He later modified its definition in the Trans. Camb. Phil. Soc. 18 (1903), pp. 107- 

 135, and still later redefined it in "The Periodogram and its Optical Analogy," Proc. Roy. 

 Soc, London, Ser. A, 77 (1906) pp. 136-140. In its final form the periodogram is equiva- 

 lent to 5w(/), where w{j) is the power spectrum defined in Section 2.1, plotted as a func- 

 tion of the period T = {2irf)~^. 



