MATIIEMAriCAL AXALV^IS OF RAX DOM NOISE 129 



Using the representation (2.8-6) we see 



2PVx cos pt = Pj^ f,Jcos (co„,/ -{- pi - v^«) + cos {o)„J - pt - ^„0] 

 1 



For the moment, we take p = l-wfL^J. The terms pertaining to frequency 

 fn = iiAf are those for which 



Wm + /> = 27r/n \(J^m — P \ = 2irfn 



m -\- r = n \ m — r\ = n 



m = n — r m = r ± n 



where only positive values of m are to be taken: If n > r, then m \s n — r 

 or r + n. If n < r, then m is r — n or r + n. In either case the values 

 of m are \n — r\ and « + r. The terms of frequency /„ in 2PTV cos ^/ 

 are therefore 



PC\n-T\ cos (2x/„/ — (p\n-r\) -\- PCn+r COS {IrfJ — iPn+r) 



and the mean square value of this expression, the average being taken over 

 the ip's, is 



- (c5„-r| + cl+r) = P^ Af[w{f\n-r\) + w{fn+r)] 



where fp denotes ^/27r. 



By combining this with the expression (4.5-5) which arises from V^ 

 we see that the continuous portion WdJ) of the power spectrum of / is 



Wcif) = a~P\uif - /,) + w{f+fp)] 



•+" _ (4.5-13) 



dx 



/-l-oo 

 w(x)w{f — x) 

 •00 



where ■zc'(— /) has the same value as «'(/). 



Equation (4.5-13) has been used to compute Wc(f) as shown in Fig. 8. 

 The input noise is assumed to be uniform over a band of width j8 centered at 

 fp , cf . Filter c, Appendix C. By noting the area under the low frequency 

 portion of the spectrum we find 



Wcif) df = a'fiwoiP' + /3wo) 

 Jo 



Since the mean square value of the input W is i/'o = |Swo , it is seen that 

 this equation agrees with the expression (4.1-15) for the mean square value 

 of Iff , the low frequency current, excluding the d.c. If audio frequency 



