MATHEMATICAL ANALYSIS OF RANDOM NOISE 143 



When the input to the square law device 



/ = aV^ (4.1-1) 



consists of noise only, so that 1' = I'a- , the correlation function for I is 



^(r) = a'[^l + 2^p;] (4.10-1) 



where \pT is the correlation function of Vx . This may be compared with 

 equation (3.9-7). \Mien V is general, 



^(r) = ave. /(/)/(/ + t) 



= Sive. a' V\t)V\t + t) 



2 s/ r- «; • . Amfiiv?. . . (4.10-2) 



= a X Coeflhcient of — - — — — ni power series expansion 



of ch. f. of V{t),V{t + t) 



where we have used a known property of the characteristic function. An 

 expression for the ch. f., denoted by g{n, v, r), is given by (4.8-4). For 

 example, when V consists of a sine wave plus noise, (4.1-13), the ch. f. is 

 obtainable from (4.9-3). Hence, 



ii" if 

 ^(t) = Coeff. of — ^ in expansion of 

 4 



(4.10-3) 



a Jo{P\/h- -\- v^ -\- 2uv cos pr) 

 X exp — y {u' + v^) — \prUV 



= a- [^ + rPoj + J cos 2pr + 2PVr COS pr + 24^1 



The first two terms give the dc and second harmonic. The last two terms 

 may be used to compute Wdf) as given by (4.5-13). 



Expressions (4.10-1) and (4.10-3) are special cases of results obtained by 

 Middleton who has studied the general theory of the quadratic rectifier by 

 using the Van Vleck-North method, described in Section 4.7. 



As an example to which the theory of Section 4.9 may be applied we con- 

 sider the sine wave plus noise, (4.1-13), to be applied to the f-law rectifier 



7 = 0, T' < 



(4.10-4) 

 7 = Y\ V > 



From the table in Appendix 4.4 it is seen that 



F(hi) = T(p + \)({u)""' 



