132 BELL SYSTEM TECHNICAL JOURNAL 



4.7 Linear Detection of Noise — The Van Vleck-North Method 



The method due to Van Vleck and North will be illustrated by using it 

 to determine the output power spectrum of a linear detector when the input 

 consists of noise alone. 



The linear detector is specified by 



^« - \i-(o, Y(o > 0, (■'■'-'^ 



which may be obtained from (4.2-1) by setting a equal to one, and the input 

 voltage is 



V(t) = ]\-(t) (4.7-2) 



where VN(t) is a noise voltage whose correlation function is i/'(t) and whose 

 power spectrum is w(f). 



The correlation function ^(r) is the average value of I(t)I(t + r). This 

 is the same as the average value of the function 



r/T' T- ^ /^ i^'2, when both Vi, V2 > /, - -.n 



^(''■'^'=\0, all other T's, ^^'''^^ 



where we have set 



T'l = y(t) 



V2 = V(t + r) 



The two-dimensional distribution of T'l and V2 is given by (3.2-4), and 

 from this it follows that the average value of any function F(Vi , F2) is 



C '"'' C '''' T[Mw "^p [-2W1 (^» '^' + ^» ''' - '-*' '- '^=>] 



(4.7-4) 

 where 



\M\=xkl-^l.l. 



For the Hnear rectifier case, where F{Vi, V2) is given by (4.7-3), the 

 integral is 



\M\~"'^l dVij^ dV2V,V2exp[--^{hVl+^PoVl-24^rViV2)j 



= ^([.2 -.;]- + ., cos- [^']) 



