142 BELL SYSTEM TECHNICAL JOURNAL 



We again take k = Z and consider Wi , m^ , niz to be one, and mi , • • • my 

 to be zero, corresponding to the modulation product np ± pi ± p2 dz ps . 

 By making the same sort of approximations as Bennett does we find 



vl„,i,i,i,o.o...o = — / F{tu)Jn{Pu)u e ^'^ du 



It 8 J c 



PiP^Psj 



= 4 ^'nZ 



When any other modulation product of the form np ± p^ ± pr., ± prs is 

 considered we get a similar expression in which P1P2P3 is replaced by 

 PriPr^Prz • This may be done for any value of k. The result indicates 

 that hnk , and consequently also the (n, k) terms in the double series 

 (4.9-10) and (4.9-12) for "^dr) and Wdf), are to be associated with the 

 modulation products of order {n, k), the n referring to the signal and the k 

 to the noise components. 



We now may state a theorem due to Middleton regarding the total power 

 in the modulation products of a given order. For a given non-linear device 

 (i.e. F(iu) is given), the total power which would be dissipated by all of the 

 modulation products which are of order {n, k) if / were to flow through a 

 resistance of one ohm is 



*.,(0)=!^';,L='4pl* (4.9-19) 



The important feature of this expression is that it depends only on the r.m.s. 

 value of T'.v and on F{iu). It depends not at all upon the s'pectral dis- 

 tribution of the noise power in the input. 



The proof of (4.9-19) is based en the relation 



^nkiO) = f Wnkif) df 



Jo 



between the total power dissipated by all the (n, k) order products and the 

 corresponding correlation function obtained from (4.9-7). 



This theorem has been used by Middleton to show that when the input 

 is confined to a relatively narrow frequency band, so that the output spec- 

 trum consists of bands, the power in each band depends only on V^ and not 

 on the spectrum of IV • 



4.10 Miscellaneous Results Obtained by Correlation Function 



Method 



In this section a number of results which may be obtained from the theory 

 given in the sections following 4.6 are given. 



