RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 105 



symmetrical about/,, and occupies the range/, — fa to/o +/a and an ideal 

 low pass iilter cutting ofif at / = /<, is used in the rectifier output, the 

 total noise output from beats between signal and noise is 



]]\„ =^ 2 r u,,Xf) df = ^^ e-''-''''"[hin\/2Wn) + Ii{W,/2W„)f (39) 



Jo -iTT 



Next we shall calculate the spectrum of the energy resulting from beats 

 between individual noise components. We write 



Ann — 2 ^flO- •■010---010-..0 



^a r ^^/o(Poz)/o(Pis)- • -JliPrZ)'- -MPsZ)-' 'MPxZ) 



T J c 



dz 



= —^ — / Jo{PoZ)e " dz 

 Itt Jq 



(40) 



2V2TrWn \2' ' Wn 



iVlirWn 



To find the resulting spectrum Wnnij)df produced at / by the resultant of 

 all such components, we note that we may sum over all components by 

 beating each component of the primary band with the frequency / above it 

 and adding the resultant power values. The result is 



'^'n„(/) = tV e-'''"'^ll{Wj2Wr.) f w{\)w{\ + f) d\ (41) 

 47rlr„ Jo 



In the particular case of a flat band of energ\^ extending from /i to /z , 



(42) 



[ w{\)wi\ +f)d\= r ,,^\,. d\ = ^) ^\ / wi 



Jo Jfi (72 — jir (72 — jir 



< / < /2 - /l 



47r (/2 - fiY (43) 



<f Kfo-fi 

 The total mean power of this type lying in the band to fb is 



TT' rf\ - f^' .;, r /Wf - «^^^"(/2 ~/l - fb/2)fb -W,/Wn 



Jo 47r(72 — 7i) (44) 



lUW>/2Wn) 



