MATHEMATICAL AX A LYSIS OF RAX DOM XOISE 125 



been discovered independently and at about the same time, by several 

 workers. In a paper read before the I.R.E., Jan. 28, 1944, D. O. North 

 described results obtained by using the correlation function. J. H. Van 

 Meek and D. Middleton have been using the two variations of the method 

 which we shall describe in Sections 4.7 and 4.8, since early in 1943. A 

 primitive form of the method of Section 4.8 had been used by A. D. Fowler 

 and the writer in some unpublished material written in 1942. Recently, 

 I have learned that a method similar to the one used by Fowler and myself 

 had already been used by Kurt Franz in 1941. 



The correlation function method avoids the problem of counting the 

 modulation products. However, in some cases it becomes rather unwieldy. 

 Probably it is best to have both methods in mind when investigating any 

 particular problem. The direct method will be illustrated by applying it 

 to the square law detector. Two approaches to the correlation function 

 method will then be described and applied to examples. 



4.5 Noise Through Square Law-Device 



Probably the most direct method of obtaining the power spectrum W(f) 

 of /, where 



/ = aV\ (4.1-1) 



V being a noise voltage, is to square the expression 



M 

 V = Vif = ^ C,n cos (oJm t — <Pm) (2.8-6) 



1 

 in which cl, is 2w{fm)Af, w™ = 27r/,„ ,/,„ = w;A/and (pi ,<f2, • • • (f.xt are random 

 phase angles. 



Considerable simplification of the algebra results when we replace the 

 representation (2.8-6) by 



V^ = \J: C.J""''-''- (4.5-1) 



Here we have added a term Co/2 so as to not have any gaps in the summation 

 and have introduced the definitions 



C—m Cm 



iP-m = —^m (4.5-2) 



a = 27rA/ 



^* "Die tjbertragung von Rauschspannung iiber den linearen Gleichrichter," Hochfr. 

 u. Elektroakust., June 1941. Other articles by Franz are (I am indebted to Dr. North 

 for the following references) "Beitrage zur Berechnung des Verhaltnisses von Signal 

 spannung zu Rauschspannung am Ausgang von Empfangern", E.X.T., 17, 215, 1940 and 

 19, 285, 1942. "Die Amphtuden von Geriiuschspannungen", E.X.T., 19, 166, 1942. 

 The May 1944 (p. 237), issue of the Wireless Engineer contains an abstract of "The In- 

 fluence of Carrier Waves on the Noise on the Far Side of Amplitude-Limiters and Linear 

 Rectifiers" by Franz and Vellat, E.N.T., Vol. 20, pp. 183-189 (Aug. 1943). 



