MATHEMATICAL AX A LYSIS OF RANDOM NOISE 147 



Ragazzini's formula is quite accurate when the index of modulation r is 

 small, especially when y — Q'/{2\{/o) is large. To show this we put r = 

 in (4.10-13) and obtain 



Wcif) ^ 



T^^Q- + 2^o) L 



_ (4.10-15) 

 + / w{x)w{f — x) dx 



where fq = q/{2ir). This is to be compared with the low frequency por- 

 tion of ITc(/j obtained by specializing (4.10-8) to obtain the output power 

 spectrum of a linear rectifier when the input consists of a sine wave plus 

 noise. The leading terms in (4.10-8) give 



Wcif) = hlMf, -f) + ^(A +/)] 



•+« (4.10-16) 



o 1 f^ 



+ /?o2 -. I w{x)w{f — x) dx 



4 J— 00 



The values of the /?'s appropriate to a linear rectifier are obtained by set- 

 ting V = 1 in (4.10-5) and noticing that Q now plays the role of P. 



hn = \(^^'\F^{h2;-y) 



y = Q'/(2h) 



Incidentally, the first approximation to the output of a Hnear rectifier 

 given by (4.10-16) is interesting in its own right. Fig. 9 shows the low fre- 

 quency portion of Wdf) as computed from (4.10-16) when the input noise 

 is uniformly distributed over a narrow frequency band of width I3,fq being 

 the mid-band frequency, //n and //02 may be obtained from the curves 

 shown in Fig. 10. In these figures P and .v replace Q and y of (4.10-17) in 

 order to keep the notation the same as in Fig. 8 for the square law device. 

 These curves may also be obtained from equations {33) to (43) of Bennett's 

 paper. 



The following values are useful for our comparison. 



When X = When x is large 



//n = hn = I/tt (4.10-18) 



A02 = (27n/'o)~'^' //02 = l/iirQ). 



The values for large x are obtained from the asymptotic expansion (45 — 3) 

 given in Appendix 45. 



