MATHEMATICAL ANALYSIS OF RAX DOM XOISE 



113 



4.2 Low Frequency Output of a Linear Rectifier 

 In the case of the hnear rectifier 



I 0, T^ < 

 \aV, V > 



the low frequenc}' output current, assuming no audio frequency filter, is 



aR 



(4.2-1) 



ia = 



(4.2-2) 



This formula, like its analogue (4.1-6) for the square law device, assumes 

 that the applied signal and noise lie within a relatively narrow band. It 

 may be used to compute the probability density and statistical properties 

 of 1 1( when the corresponding information regarding the envelope R of the 

 applied voltage is known. 



The truth of (4.2-2) may be seen by considering the output /. It con- 

 sists of the positive halves of the oscillations of aT'. The envelope of / is 

 the same as that of a]'. However, the area under the loops of / is only about 

 I/tt of the area under aR, this being the ratio of the area under a loop of 

 sin X to the area of a rectangle of unit height and length 27r. From the 

 low frequency point of view these loops of / merge into a current \yhich 

 varies as aR/ir. 



When 1' is a sine wave plus noise, 



V = Fjv + P cos pt 



(4.1-13) 



the average value of /^^is^ 



'-=='-(£)"' .'.(-1--I.) 



(£)"■.-"[(. + •)'.© 



(l+a;)7o(^j + xA(| 



(4.2-3) 



where h , 7i are Bessel functions of imaginary argument and 

 _ P _ ave. sine wave power 



2h 



ave. noise power 



(4.2-4) 



*^ This result was discovered independently by several investigators, among whom we 

 may mention W. R. Bennett and D. O. North. The latter has appUed it to noise measure- 

 ment work. He has found that the diode detector, when adapted to noise metering, is a 

 great improvement over the thermocouple, and has used noise meters of this type satis- 

 factorily since 1940. See D. O. North, "The Modification of Noise by Certain Non- 

 Linear Devices", Paper read before I.R.E., Jan. 28, 1944. 



