TIME DIVISION MULTIPLEX SYSTEMS 



211 



F^{t) = 



0, - (2 N - x)T/2 N < t < - X T/2 N ^ 

 A, - xT/2N <t < xT/2N 

 0,xT/2N<t< {2N - x) T/2 N 

 -A, {2N-x)T/2N<t<{2N-\-x)T/2N^ 



(36) 



The relation of Fj(f) to Fi{t) is as specified by (15). By the usual Fourier 

 series expansion, we then find 



F^(f) = Z TTT-^^ Sin (2^ +/)^^ cos {2m + 1)9^2 (37) 



t^o {2m + l)ir 



2N 



where q = 2Tr/T. In the limiting case in which the duration of contact 

 time is made small compared with the interval between contacts, the 

 expression (37) combined with (15) gives the general result: 



Fj{t) = -^ Zl cos 



' {2m + l)qt _ {2m + l){j - l)7r " 

 2 N 



(38) 



By comparing this function with (35), setting v = q/2, we note that plus- 

 and-minus switching with small contact time can be made to give freedom 

 from crosstalk and economy of band width in a. time division multiplex 

 system provided that we select the sidebands on N/2 adjacent odd harmonics 

 of q/2, beginning with m = rN/2., i.e., the {rN +1)^* harmonic of q/2, where r 

 may be zero or any positive integer. It is necessary that even harmonics of 

 q/2 be suppressed, which will be the case if the positive and negative 

 switching contacts are identical except for change in sign. 



Vestigial Sideband Systems 



It is seen that the use of a commutator results in the production of a 

 large number of sidebands, and that for economy of bandwidth in the trans- 

 mission medium it is desirable to select only a definite group by filtering. 

 The requirements on the filter are not easily met since there should be linear 

 phase shift and uniform attenuation throughout the range of frequencies 

 occupied by the desired sidebands and a high degree of suppression of all 

 other frequencies. The requirements may be simplified somewhat by using 

 the principle of vestigial sideband transmission which allows the attenuation 

 to increase gradually within the desired band and compensates for energy 

 removed from wanted sidebands by allowing a corresponding amount of 

 energy to be transmitted from adjacent sidebands which would normally be 

 suppressed. Linear phase shift is required for all frequencies at which 

 appreciable transmission exists. 



