200 BELL SYSTEM TECHNICAL JOURNAL 



sideband philosophy of signal transmission, illuminates the manner in 

 which departures from ideal amplitude and phase characteristics cause 

 crosstalk between the several message channels. It further leads directly 

 to other physical methods for producing and detecting a transmitted signal 

 identical with the essential components derived in time division or switching 

 processes. 



A first step in the theoretical solution of the problem was taken by 

 Dr. J. R. Carson, who, in an unpublished memorandum of May 25, 1920, 

 derived quantitative relations between band width and interchannel inter- 

 ference in time division multiplex transmission. Applying Fourier series 

 analysis to on-and-o£f switching, he showed that if the transmission medium 

 had constant attenuation and linear phase shift for all frequencies below 

 cutoff and no transmission of frequencies above the cutoff, the band width 

 required for satisfactory multichannel telephony would be much wider than 

 needed in conventional carrier methods. A further step was taken by 

 Dr. H. Nyquist, who, in unpublished memoranda of August 24, 1936 and 

 November 12, 1936, showed that the width of band necessary may be 

 reduced by providing a specially devised type of non-uniform transmission 

 characteristic. In the following discussion, we shall see that a similar 

 result can be obtained by control of the switching, and specific switching 

 processes will be described which allow a flat transmission band of minimum 

 width to be used. 



In order to arrive at requirements which must be imposed on the various 

 components of the system, we shall first give a theory of time division 

 multiplex transmission in which both the switching processes and the 

 transmission characteristic are completely general. Specific forms which 

 give crosstalk suppression will then be discussed and effects of small de- 

 partures estimated. 



General Theory 



We shall assume an TV-channel system with a sinusoidal signal impressed 

 on the 7*'' channel. An illustrative arrangement is shown in Fig. 1. Since 

 the system is linear, we may represent currents and voltages by complex 

 quantities with the understanding that the actual currents and voltages 

 are the real components of the expressions used. Accordingly, let the 

 signal voltage impressed on the j^^ channel be 



£;(/) = Eic"-'' (1) 



' Basic concepts used in Nyquist's analysis were included in his paper, "Certain 

 Topics in Telegraph Transmission Theory," A. I. E. E., Trans., April, 1928, pp. 617-644. 

 Mention is also there made of the equivalence of signal shaping and equalizing in effect 

 on reception of telegraph signals. 



