882 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Let the last integral in (1.7) be F{t) so that d(t) = Im F{i). Then 



e{tm + r) = 1 Re {F{t)F*{t + r) - Fit)F{t + r)} (2.3) 



where F*{t + r) is the complex conjugate of F{t + r). The autocorrela- 

 tion function of d{t) is obtained by averaging over the ensemble of the 

 noise functions (p{t) : 



Reir) = av d{t)eit + r) 



= av - Re < / dtj dt" exp [i^pit' — /) + iip{t') — i^pit)] 



■g{t - t')g{t + r - /'0[exp \-iv{t" - i - r) (2.4) 



- tV(r) + z^(^ + t)] - exp UX^" - / - r) + i^{t") 



- i<p{f + t)]|. 



Since g{t) is real, ^*(0 = g(t). The averaging process may be carried I 

 out by a method analogous to that used in Reference 4. The formula to 

 be used is 



av exp [ixp{t') - i<p(f) + ia<p(f') - ia<p{t + t)] 



= exp [— i/'o(l + a') + \pi'-t — a\pv-t" + a^v-t-r (2.5) 



+ a^t^f — a^/r + aVr'-^-r] 



where a is either — 1 or +1, and xpr is an even function of t. When (2.5) 

 is used in (2.4) a double integral for Reir) is obtained. The substitutions 



X = t - t', 



y =^t-V T -t", (2.6) ) 



i?i, = ypT+x-v — ^T+x — ^T+x — 4'r-y + ^r 



convert the double integral into 



Reir) = I Re r dx r dyg(x)e-''''-'''>^'^^'^ ^ ^ 



Z J-00 J-« (2.7) 



The symbol R^ is chosen to agree as closely as possible with the notation 

 of Reference 4. There R^ w^as the autocorrelation of the random func- 

 tion, vii), where v{l + T) = ^(0 - ^{t + T), T being the echo delay. 

 Here, R^ is the average value of the product, 



[<p{t) - <p{t + y)] y{t + t) - <f{t + r + x)] 



which becomes the autocorrelation function of v{t) when y = x = T. 



