202 BELL SYSTEM TECHNICAL JOURNAL 



The wave (3) consists of the output of the circuit of Fig. 1 with all frequen- 

 cies shifted by a constant amount u radians per second; various means of 

 accomplishing this result in the switching process will be discussed later. 

 It is sufficient to point out here that such a shift in frequency is often de- 

 sirable for optimum utilization of the transmission medium. Combining (2) 

 and (3), we then have: 



2 m=0 



It is clear from (5) that the result of the switching process is the produc- 

 tion of upper and lower side frequencies from the signal on each harmonic 

 of the switching frequency. It is also evident that if more than one signal 

 component is superimposed, the resulting side frequencies constitute side- 

 bands of the same nature as used in amplitude modulation systems. A 

 significant difference between time division and amplitude modulation 

 appears in that in the latter only one sideband or at most one pair of side- 

 bands is transmitted, while the essential character of time division depends 

 on the transmission of a plurality of sidebands. Thus if one pair of side- 

 bands were selected from the output (5) by filtering, the time division 

 process would merely be a particular way of generating the sidebands 

 required in an amplitude modulation system. 



The next step in a time division system is the transmission of the wave (5) 

 over a line. The properties of the line in general may be specified by a 

 complex transfer impedance, which we may express here by the ratio of 

 open-circuit output voltage to input current: 



Er/Is = Z{u,) (6) 



The result of applying the wave (5) to the line is then the open-circuit 

 voltage: 



+ f i: A^,-Z*[i{v + mq - ^. )],«"+'"«-";) '+^-«"u- 



i m=0 



In the above we have adopted the notation Z*(ico) to represent the con- 

 jugate of Z(io}) and have made use of the fact that the response of a network 

 to the applied wave e~""' is the conjugate of the response to e^" . 



At the receiving end another switching process takes place synchronously 

 with that at the transmitting end. We shall assume that the switching 

 process between the k^^ channel and the line is represented by the relation 



/,,(/) = Gk{t)Eri{t), (8) 



