102 BELL SYSTEM TECHNICAL JOURNAL 



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i ^ ' ' ' ' ' ' ' ' ^TOomi"' 



.„,^ COS wqXo cos m\X\ 



where 



mo=0 mi=0 iBjv=0 (18) 



• • • COS DIn Xn 



Xk = pkt, k = 0,\,2,"' N (19) 



= '"° "'";,''/ '"'^ r dxo f dxv 



■k"'^^ Jo Jq 



j I cos nio Xo COS nii xi- • • cos W/vXat dxif (20) 



Jo 



'■mo»n- • -TOAf 



(21) 



The response of the rectifier is thus seen to consist of ah orders of modula- 

 tion products of signal and noise. In a typical case of interest the band of 

 input frequencies is relatively narrow and centered about a high frequency 

 while the output band includes only low frequencies. In such a case the 

 important components in the output are the beats between signal and noise 

 components and between noise components. The d-c component is present 

 in the output only if the pass band of the system actually includes zero 

 frequency; we have already computed its value in Section I, but we will 

 derive it again by the method used here as a check. 



The amplitude of the d-c component is in fact: 



Ooo 



a / „=i , {22) 



...0 = - TT Zi dz, 



2ir J c 2 



on substitution of the expression for E in the integral representation of 7, 

 substituting the result in (20) and interchanging the order of integration. 

 When N is large, P„ is small, hence the principal contribution to the integral 

 occurs near small values of z, where Jo{Pnz) is nearly equal to unity, since 

 the product of a large number of factors, all less than unity, will be small 

 indeed unless each factor is only slightly less than unity. We therefore 

 replace Jo{Pnz) by a function which coincides with it near z — Q and goes 

 rapidly to zero as we depart from this region. Such an approximation 

 (Laplace's process ) is 



MPnz) = e-'"^'"" (23) 



2 Watson, "Theory of Bessel Functions," p. 421. 



