206 BELL SYSTEM TECHNICAL JOURNAL 



If the attenuation of the line is constant throughout the range co/ to Mq + 

 CO; and all frequencies above the latter value are suppressed, (21) becomes 





1 + 2 



I J—- ) cos {] - k)2nnr N (22) 



m=i \ mx-K/N J J 



The crosstalk ratio or ratio of amplitude of signal received in the k^^ channel 

 to that received in the j^^ channel when signal is transmitted in the y*^ 

 channel is, therefore, 



J ^ 2 I; /sin ««^y ^^^ 



Yh 



I > 2 Y" / sin ^TT x/N V 

 m=i \ m-Kx/N ) 



Results of calculations made for a 10-channel system from (23) for 

 .V — 1 and X = .5, corresponding to no lost time and half lost time respec- 

 tively in switching are shown in Fig. 2. It may be noted that adjacent 

 channel crosstalk with half lost time is equivalent to alternate channel 

 crosstalk with no lost time. Examination of the curves reveals a number 

 of significant facts, among which are: 



1. Crosstalk is quite imperfectly suppressed when the band width of the 

 line is smaller than the theoretical minimum — the width of one sideband 

 multiplied by the number of channels. 



2. As the band width of the line is increased above the theoretical mini- 

 mum, improvement in crosstalk suppression increases slowly, so that in 

 general the use of frequency range on the line is uneconomical compared 

 with other systems. For example, with no lost time in switching, the band 

 width of the line must be increased tenfold to suppress adjacent channel 

 crosstalk by 40 dh. This conclusion is, however, to be qualified as follows: 



3. When the duration of contact is decreased (less of the available channel 

 time used) definite optimum transmission band widths appear which give 

 better crosstalk suppression than bands somewhat wider or narrower. This 

 suggests the possibility of critical phase relations existing between the con- 

 tributions from the various sidebands such that if the right number having 

 proper amplitudes and phases can be combined, complete suppression of 

 crosstalk may occur even when the transmitted band width is finite. 



When .r, the fraction of contact time used, is made to approach zero, 

 the limit of the amplitude factor (18) for the typical harmonic of the switch- 

 ing function is A^x = 2Ax/N, which is independent of m. This is con- 

 sistent with the known fact that a wave consisting of periodically repeated 

 sharp pulses is composed of a large number of harmonics of nearly equal 



