TIME DIVISION MULTIPLEX SYSTEMS 203 



where IrkiO is the current received in the k^^ channel and Gk(t) is a periodic 

 function of time with fundamental frequency q. It is understood that j 

 and k may be any two of the N channels. We shall express Gk(t) in a 

 manner analogous to the corresponding switching function at the trans- 

 mitter, i.e., 



00 



Gkit) = E Bnk cos [{u + nq)t - $„,] (9) 



n=0 



Combining (7), (8), and (9), we find 



7-1 00 QO 



Irk(t) = ~ Yl ^ AmjBnkZ[i{v + mq + OJy)] 



4 »n=0 n=0 



77 °° °o 



+ "T 2 2^ AmiBnkZ*[i{v -\- mq - co,)] 



4 jn=0 n=0' 

 / — jl2y+(TO+n)g— Mj]<+i(9mj+*nfc) I i\.in—m)g+wj]t+i(Bm] — ^nk^'\ (ifW 



The received wave thus consists of a doubly infinite set of side frequencies 

 involving harmonics of q. It is, however, possible to set up conditions 

 under which the original signal may be selected and the frequencies involving 

 the switching rate may be suppressed by filtering. If j' == 0, such separation 

 is possible provided 



o^i<q/2, (11) 



for it then follows that a low-pass filter with cutoff frequency at q/2 will 

 not pass any of the components with frequencies dependent on q. The 

 condition (11) follows from the fact that the lowest frequency of (10) 

 dependent on g is 9 — coy, and hence we must make q — coj > co; in order to 

 separate o^j from q — co/. In other words the sidebands on adjacent har- 

 monics must not overlap. If j' > 0, the condition (11) also suffices as far 

 as suppression of terms dependent on q are concerned, but an additional 

 condition is required to suppress frequencies dependent on v in the special 

 case in which v < q/2, i.e., the case of v less than the maximum allowable 

 value of coy. For in the latter case the frequency 2v -{- (m -\- n)q — coy is 

 less than q — coy in the special case of m — n = 0. The additional condition 

 needed is evidently either /loy = or Bqu = 0. If i' = or if ^^ > q/2, 

 this condition is unnecessary. 



Assuming then that (11) is fulfilled, and that a low-pass filter with cutoff 

 at q/2 is inserted in the output of each channel, we calculate for the typical 

 channel output current : 



hk{t) = Fy,£y/"'^ (12) 



