MATHEMATICAL ANALYSIS OF RANDOM NOISE 145 



k increase. The low frequency portion of the continuous portion of the 

 output power spectrum is then, from (4.9-12), 



Wcif) = ^^hl.G^if) + ^,/4G4(/) + ••• 



+ j^j hUCrif - /o) + Gi(/ + /o)] + |, hUC^if - /o) (4.10-8) 



+ Gz(f + /o)] + |j hUG^if - 2/o) + G2(f + 2/o)] + • • . 



From Table 2 of Appendix 4C we may pick out the low frequency portions of 

 the G's. It must be remembered that Gm(x) is an even function of x and 

 thatO </«/o. 



As an example we take the input noise Vy to have the same w(f) and 

 \P{t) as Filter a, the normal law filter, of Appendix 4C, so that 



and assume that the sine wave signal is at the middle of the band, giving 

 p = 2t/o . Thus, from (4.10-8), for low frequencies and the normal law 

 distribution of the input noise power, 



1 .2 ,2-/2/4(72 , ^ ,2 4 -/2/8<r2 





I 72 ,2 -f^lia- 1 



Although we have been speaking of the I'-law rectifier, equation (4.10-9) 

 gives the low frequency portion of TFc(/), corresponding to a normal law 

 noise power, for any non-hnear device provided the proper /7„;t's are inserted. 



When we set v equal to one in the expression (4.10-5) for //„a- we may ob- 

 tain the results given by Bennett. Middleton has studied the output of a 

 biased linear rectifier, when the input consists of a sine wave plus noise, and 

 also the special case of the unbiased linear rectifier. He has computed the 

 output for a wide range of the ratios P'/'Ao , B''/\po where B is the bias. In 

 order to cover the entire range he had to derive two series for the corre- 

 sponding linkS, each series being suitable for its particular portion of the 

 range. 



