140 BELL SYSTEM TECHNICAL JOURNAL 



Incidentally, these expressions for the amplitudes follow almost at once from 

 the direct method of solution. This will be shown in connection with equa- 

 tion (4.9-17). 



Since the correlation function "^c(t) for the continuous portion Wdf) of 

 the power spectrum for / is given by 



^e(r) = ^(r) - ^„(t), (4.8-11) 



we also have 



00 00 A 



^c(t) = Z Z x^ i^rhlken COS npr (4.9-10) 



71=0 fc=i «! 



Wlien this is substituted in 



W 

 we obtain 



(/) = 4 [ ^o(t) cos lirfr dr (4.9-11) 



Jo 



where 



Gicif) = [ 'Ar COS lirfr dr (4.9-13) 



Jo 



is the function studied in Appendix 4C. Gkif) is an even function of/. The 

 double series (4.9-12) for Wc looks rather formidable. However, when we 

 are interested in a particular portion of the frequency spectrum often only 

 a few terms of the series are needed. 



It has been mentioned above that the direct method of obtaining the out- 

 put power spectrum is closely related to the equations just derived. We 

 now study this relation. 



We start with the following result from modulation theory : Let the 

 voltage 



V = Po cos .vo + Pi cos .Vi + • • • -f- Pn cos Xn 



(4.9-14) 

 Xk = pj, k = 0,\, ■•• N, 



where the ^/,'s are incommensurable, be applied to the device (4A-1). The 

 output current is 



05 00 



i = Z_* ••• 2^ 2-^mo--m^^mo 



mo=0 mfj=0 (4.9-15) 



• • ' Cmjv COS moXo cos ftliXi • • • cos MffXN 



^° Bennett and Rice, "Note on Methods of Computing Modulation Products," Phil. 

 Mag. S.7, V. 18, pp. 422-424, Sept. 1934, and Bennett's paper cited in Section 4.0. 



