120 BELL SYSTEM TECHNICAL JOURNAL 



and contains the sine wave frequency fq. The noise current may be re- 

 solved into two components, one "in phase" and the other "in quadrature" 

 with Q cos qt. Using the representation (2.8-6) of reference A and proceed- 

 ing as in Section 3.7 of that paper: 



Is = 2 ^n COS (cOn/ " <^n) (3.1) 



n=l 



M 



= X) ^n COS [(cOn — q)t — (pn + qt] 

 n=l 



= Ic COS qt — Is sin qt (3.2) 



where 



^ Cn COS [(cOn — q)t — ipn] 



n=l 



0-^) 



h = 2 ^n sin [{o3n — q)t 



COn = 2ir/„, fn = wA/, c„ = 2w{fn)Af 



w{f) denotes the power spectrum of In and the <^n's are random variables 

 distributed uniformly over the interval (0, 27r). 

 The total current / may be written as 



I = Q cos qt -\- In 



= {Q + Ic) cos qt — Is sin qt 



. (3.4) 



= R cos 6 cos qt — R sm d sin qt 



= R cos (9/ + d) 



where we have introduced the envelope function R and the phase angle d 

 by means of 



RcosB = Q+ le 



Rsmd = Is 



Since /c and /, are functions of / whose variations are relatively slow in 

 comparison with those of cos qt, the same is true of R and (usually) 6. 



A graphical illustration of equations (3.4) and (3.5) which is often used is 

 shown in Fig. 4. 



In accordance with the usual convention used in alternating current 

 theory, the vector OQ is supposed to be rotating about the origin with 

 angular velocity q. If In happened to have the frequency q/lr, its vector 

 representation QT would be fixed relative to OQ. In general, however, the 



