MATHEMATICAL ANALYSIS OF RANDOM NOISE 137 



where the phase angle tp cannot be determined from (2.2-3) since it does not 

 affect the average power. 



Consider the correlation function for V{i) = Vs{t) + TatC/) given by 

 (4.8-2). It is 



.7" 



Limit il f Vs(t)Vs(t + r)dt+ [ VM)VAt + r) dt 



+ jf VAi)Vs(t-h T)dt + f^ VAi)VAi + r)dt\ 



(4.8-8) 



Since Vs(t) and T'iv(0 are unrelated the contributions of the second and 

 third integrals vanish leaving us with the result 



Correlation function of T'(/) = Correlation function of Fs(/) 



+ Correlation function of T a'(/). 



Now as T -^ =0 the correlation function of 1^(0 becomes zero while that of 

 Vs{t) becomes of the type (2.2-3) given above. Hence the correlation func- 

 tion of the regular voltage Vs{t) may be obtained from V{t) by letting r — > cc 

 and picking out the non-vanishing terms. Although we have been speaking 

 of V{t), the same results hold for I(t) and this process may be used to pick 

 out those parts of ^(r) which correspond to the dc and periodic components 

 of I(t). Thus, if we look at (4.8-7) we see that as r -^ cc , i//^ — > 0, while the 

 gs {u, V, t) corresponding to Vs{t) given by (4.8-5) remains unchanged in 

 general magnitude. This last statement may be hard to see, but examina- 

 tion of the cases discussed later show that it is true, at least for these cases. 

 Thus the portion of ^(r) corresponding to the dc and periodic components 

 of /(/) is, setting i/'^ = in (4.8-7), 



^^{r) = ^J F{m)e-'^'>""'' du [ F{iv)e-'^'"'''" gs{u, v, r) dv (4.8-10) 

 47r" J c *' c 



where the subscript =o indicates that ^oo(t) is that part of ^(t) which does 

 not vanish as t -^ co . 



We may write (4.8-9), when applied to I{t), as 



^(r) = M^«(t) -I- M^e(T) (4.8-11) 



where ^c{t) is the correlation function of the "continuous" portion of the 

 power spectrum of /(/). 



Incidentally, the separation of ^{t) into the two parts shown in (4.8-11) 

 may be avoided if one is willing to use the 8(1) functions in order to interpret 

 the integral in (4.6-1) as explained in Section 2.2. This method gives the 

 proper dc and sinusoidal components even though (4.6-1) does not con- 

 verge (because of the presence of the terms leading to ■^oo(t)). 



