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BELL SYSTEM TECHNICAL JOURNAL 



K{ 



^+2 {z[i{v-\-mq-\-wi)\-\-z*[i{v + mq-u,)]), j = k 



E {z[i{v + mq + ^.)],-^-(i-.)(2™-2„+i)Wiv 



+ z*[i{v -\-mq — cOj)]e 



x{j-k)(2m-2n+\)irlN\ 



(43) 



0, i ^ -^J 



where i^T is a complex constant of proportionality. The case oij— k which 

 gives transmission within the channel contains a variation with signal 

 frequency caused by the summation of the departures from ideal transmis- 

 sion at the N sideband frequencies. This term presumably will be unimpor- 

 tant if the transmission characteristic is sufficiently good to meet crosstalk 

 requirements; hence we may neglect the z and z* terms in the case of j = k 

 and write the ratio of interference to desired signal as: 



Cjk — 



F,v 



N 



[z[i{v -j- mq -^ cojjje 



+ z*[i{v -^ mq - ^.)]g'-0-«C2™-2n+l)WiV) (44) 



The crosstalk ratio will in general vary with the signal frequency. The 

 requirement would logically be based on the total interference power 

 weighted in accordance with the interfering effect at individual frequencies. 

 Thus we might set 



Xjk = Sj I Wjkicx}]) \Cikfduj, 



(45) 



where Xjk represents the weighted interference power received in the k^^ 

 channel when a reference signal wave of mean power 5, is applied to the^*^ 

 channel. The limits of integration oja and co^ are the lowest and highest 

 signal frequencies used. The function Wjk{ooj) represents the proper 

 weighting with frequency of the interference and takes into account the 

 distribution of the interfering signal and the relative importance of the 

 different interfering frequencies. 



Equation (45) is sufficient for computation of interchannel interference 

 introduced by the line when the transmission characteristics of the line are 

 known. A more valuable result, however, would be the expression of the 

 required line characteristics in terms of the allowable interference. In 

 general this would require some specification of the nature of the departures 

 from the ideal characteristic. Except perhaps for systems with very few 

 channels, it seems reasonable to assume that the departures are distributed 

 fairly uniformly throughout the frequency range transmitted by the line, 



