108 BELL SYSTEM TECHNICAL JOURNAL 



expanding the envelope of the input wave by the binomial theorem and 

 retaining only the first two terms. The validity depends on the noise 

 amplitude being small compared with the sum of signal and noise, and hence 

 the result should agree with our solution in the neighborhood of TT'„/tT's = 0, 

 which it does. WTien TT',, U',, is small, the error is appreciable. Ragazzini's 

 result (Equation 15 of the paper) expressed in our notation is 



^•' + " "" = 7= 1 + w./w. ^'^^ 



It will be seen by comparing the limiting values for Trs/Tr„ = with that of 

 Trs/Tr„ = ^ from (49) that the intercept of the curve of Fig. 4 would be 

 50% instead of our value of 39.27%. 



The results given in the present paper have been compiled from unpub- 

 lished memoranda and notes by the author extending back as far as 1935. 

 Discussions with colleagues have been of great aid, and in particular ac- 

 knowledgment is made to Messrs. S. 0. Rice and R. Clark Jones for many 

 helpful suggestions. 



APPENDIX I 

 EVALUATION OF INTEGRAL FOR I 



Interchanging the order of integration in (9), we have 



7 = .^ r dd f e-'--'^ ""^ '''"'''^ z dz (50) 



7rv27rlKn •'0 •'0 



By substituting 2 = Po cos + n \/2lF„ , we may evaluate the second inte- 

 gral in terms of the error function, obtaining 



T = ^f dd f _ e""'(M \/2?F"n + Po COS d) du 



TT'- Jo J-P cos «/\/2Wn 



_ a -ylVn r -Pq^ cos2 ei2W„ ^Q 

 IT l-K Jo 



+ ^ [ erf (Po cos d/VWn) cos 6 

 27r Jo 



dd 



_ a y/W n g-P§/4W„ C ^-co3 29/4Jr„ ^^ 

 TT l-K Jo 



