204 BELL SYSTEM TECHNICAL JOURNAL 



where the value of Y jk is as follows: 

 Case 1, J' = 



Yik = AoiBokZH'.'i) 



4 ni = \ 



+ I Z .4„,y5„,,Z*[z-(m9 - a;;)]e'''""-*"^^ (13) 



4 m = l 



Case 2, V > 0; .4o/ or 5oa = if z' < q/2 



Yj, = \ i: A„,B„„Z[i{p + W9 + c.Or''^--*'"^-' 



4 m=0 



+ ,^ E ^I„„5.,Z*[i(»' + ^>/9 - c.iW''-'-^-"' (14) 



4 m=0 



The combination of an A^-channel time division multiplex system with 

 low-pass filters in the receiving branches is thus found to be equivalent to 

 a linear network having TV pairs of input and output terminals with the 

 transfer admittance from the 7^'' pair of input terminals to the k^^ pair of 

 output terminals given by I'y/, in (13) or (14). The transfer admittance is 

 calculated by summing the contributions of upper and lower sidebands on 

 harmonics of the switching frequency and is affected directly by the trans- 

 mitting properties of the medium at the side band frequencies. The result 

 we have obtained is of sufficient generality to include all cases we shall 

 treat in this paper. We shall now proceed to specific examples. 



On-and-Off Switching with Commutator 



When an ideal commutator is used as a switching means, the switching 

 functions for the N channels are identical except for a time displacement 

 which is the same between all pairs of consecutive channels. This condition 

 is expressed by : 



F,(t) = F,[t - (j - 1)T/N] (15) 



Thus Fi{t), the switching function for the first channel becomes a reference 

 function, F2{t) is the same except for a time delay of T/N, Fs{t) is delayed 

 by 2T/N, etc. Substitution of (15) in (3) gives the relations: 



■Ami ^^ Ami I 



(16) 

 dmi = dmi + (7- l)2mTr/N] 



If we further suppose that the commutator makes contact between the 

 typical channel and the line throughout a fraction x of the time interval 



