MATHEMATICAL ANALYSIS OF RANDOM NOISE 121 



Taking n to be one shows that the band around /„, is given by 



R 



(4.3-12) 



The statistical properties of the low frequency output and of the en- 

 velopes of the output bands may be obtained from those of R. For ex- 

 ample, the probabihty density of An(R) is of the form 



p{R)/ 



^-^ (4.3-13) 



dR 



where p{R) is the probability density of R. In this expression R is con- 

 sidered as a function of An . 



It should be noted that we have been assuming that all of the band 

 surrounding the harmonic frequency nfn is taken. Wlien we take only a 

 portion of it, presumably the statistical properties will tend to approach 

 those of a random noise current in accordance with the first statement made 

 at the beginning of this section. 



WTien we apply (4.3-11) to the square law device we have 



Ztti J 



(0+) 



2m 



_ « I?2 



When we apply (4.3-11) to the linear rectifier; 



F{iu) = — 



u 



+ 00 



J^juR) . ^oR 



where the path of integration passes under the origin. These two results 

 agree with those obtained in Section 4.1 and 4.2 from simple considerations. 

 As a final example we find the low frequency output of a biased linear 

 rectifier in terms of the envelope R of the applied voltage. From the table 

 of F{;iii) given in Appendix 4A we see that F{iu) corresponding to 



/ = 0, V <B 



I = V - B, V > B 



