538 BELL SYSTEM TECHNICAL JOURNAL 



where 



F^^ = FJ" + F,' + 2FcFs cos (dc - 6^), (156) 



FJ = Fo^ + F32 - 2FcF, cos {do - e,). (16&) 



The limits of integration of w„ are determined by the fact that, 

 CO — con in the first integral of (146) and w + co„ in the second, must 

 lie between ± coai all other frequencies are eliminated by the low 

 pass filter. 



From formula (146) and the Fourier integral energy theorem, the 

 noise power Pn is given by 



JTN = . ^rp I Pp-do) I I COo + C0„ + - CO 1 doOn 



t/0 i/oj— w^ \ ' 



+ ;r"^ I -^m^^^ I OJo + C0„ - -CO (/C0„. (176) 



4T^rj^ J-(co+co„) V ^ ^ 



Integrating with respect to co„, we have 

 Pn = ^ I cfcof [(coo + (1 + '^)co)2 + ico„2]F^2 



+ [(coo - (1 + ^)C0)'^ + |C0„2]F„2| ^ (18^,) 



where v = /x/X. 



Replacing Fp^ and Fm^ in (186) by their values as given by (156) 

 and (166), we get 



P^=-^ \ ('^O' + (1 + VY^' + \^a'){F-' + F.')do: 



+ 4-^ (1 + v)cooo^FcFs cos (0, - 0,)c?co. (196) 

 'r ^ Jo 



To evaluate (196) we make use of the formulas, derived below 



c^ + F/)do: = 1. (206) 



H''"' 





-^ rco2(F.2 + F.2)^co = XV = Ps, (216) 



wFcFs cos (^c - ^.)^co -^ as r -^ 00 . (226) 



Substitution of (206), (216), (226) in (196) gives for large values of T 



Pn = (W + coo^ + (1 + v)2X2?)co„iV2. (236) 



Here, for convenience, we have replaced N-Jtv^ of (196) by N~, so that 

 N^ of (236) may be defined and regarded as the high frequency noise 

 power level. 



