208 BELL SYSTEM TECHNICAL JOURNAL 



M = {N - l)/2, 1 

 I 

 it follows that > (25) 



Thus there exists in theory a system employing sidebands on zero fre- 

 quency and the first {N — l)/2 harmonics of the switching frequency, in 

 which multichannel transmission is possible without interchannel inter- 

 ference. Since the required condition (25) may also be written N — 

 2M + 1, an odd number of channels is obtained. Since N sidebands are 

 transmitted, the band width used is the same as the minimum required for A" 

 single sideband amplitude modulation channels on a frequency discrimina- 

 tion basis. Sidebands produced on higher harmonics in the time division 

 process must be removed by filtering. 



It is to be noted that since it is equality of the N sideband contributions 

 which is important and the amount of each contribution is determined by 

 the transmission characteristic of the line as well as the transmitting and 

 receiving switching processes, it would be theoretically possible to make up 

 for sideband irregularities by equalizing the line. However, the equaliza- 

 tion required in the line would be of "stairstep" type rather than smoothly 

 varying with frequency since an error in the value of one harmonic of the 

 switching function produces the same error throughout the entire range 

 occupied by the pair of sidebands associated with that harmonic. 



General Switching Functions with Crosstalk Suppression and 

 Minimum Band Width 



The above discussion based on the properties of a commutator has led 

 us to an ideal switching function which is, except for an unimportant 

 proportionality factor, 



(JV-l)/2 



Fj(t) = 1+2 X) cos m[qt - ij - l)27r/N], TV odd (26) 



This type of switching is approximately realizable with synchronized com- 

 mutators having contact widths very narrow in comparison with the spacing 

 between contacts. For a 3000-cycle speech band, the minimum switching 

 rate would be 6000 cycles per second. Such a speed would be difficult to 

 obtain with ordinary mechanical means but would be feasible with rotating 

 electron beams. 



The concept of combining detected contributions from a number of 

 sidebands in proper phase to give in-phase addition of desired components 

 and cancellation of unwanted ones leads to a generalization of the switching 

 processes over those possible with synchronized commutators. We note 

 that the switching functions Fj(t) of (2) and Gk{t) of (9) are analogous to 



