138 BELL SYSTEM TECHNICAL JOURNAL 



4.9 XoisE Plus Sine Wave Applied to Non-Linear Device 



In order to illustrate the characteristic function method described in 

 Section 4.8 we shall consider the case of a non-linear device specified by 



/ = J- f F{m)e''"' du (4A-1) 



Iir J c 



when V consists of a noise voltage plus a sine wave : 



Vit) = P cos pt + V^it) (4.1-13) 



As usual, F.v(/) has the power spectrum k'(/) and the correlation function 

 ;^(r). \f/(r) is often written as xj/r for the sake of shortness. Comparing 

 (4.1-13) with (4.8-2) gives 



F,(/) = P cos pt (4.9-1) 



Our first task is to compute the ch. f. gsiu, v, r) for the pair of random 

 variables VsQ) and Vs{t -\- t). We do this by using the integral (4.8-5): 



1 r 



gs{u, V, t) = Limit - / exp [luP cos pt + ivP cos p{t + r)] dt 

 r-»oc T Jo 



(4.9-2) 



= Jo{P\/u' + V- -f 2uv cos Pt) 

 where Jo is a Bessel function. The integration is performed by writing 

 u cos pt -\- V cos p (t -\- t) = (u + V cos pr) cos pt — V sin pr sin pt 



= a/w- -\- V" -{- 2nv cos />r cos {pt -f phase angle) 

 and using the integral 



Mz) =^ f 



Iir Jn 



The correlation function for (4.1-13) has also been given in Section 3.10. 



The correlation function "^(t) for /(/) may now be obtained by substi- 

 tuting the above expressions in (4.8-7) 



^(r) = A f du F{iii)e-^'^'"^"' [ dv F{iv)e 



■iTT- Jc J C 



(4.9-3) 

 e~'^'-'"'/o(P a/m^ + t)2 + 2uv cos pr) . 



^oc{t), the correlation function for the d.c. and periodic components of /, 

 may, according to (4.8-10), be obtained from this by setting 4't equal to zero. 

 When we have a particular non-linear device in mind the appropriate 

 F{m) may often be obtained from Appendix 4A. For example, F(iu) for a 

 linear rectifier is —u~'. Inserting this value in (4.9-3) gives a definite 



