TllEOHKTlCAL FUXDAMEM'ALtJ OF I'LLSE TKAAfciMlttaiON 1003 



For various values oi n ^ Amin/Am^^^ the margin for unit difference in 

 pulse amplitudes becomes 



fjL 0.2 0.3 O.l 0.-) 0.0 0.7 0.75 



,1/1 0.30 0.375 0.40 0.375 0.34 0.25 0.13 



The optimum condition is thus in the aboxe j)articular case ol)tained 

 with 11 = 0.3, with a comparatively small variation in transmission per- 

 formance for any value of n between and 0.5. 



In the above discussion of vestigial sideband transmission, modulation 

 of a carrier was assumed, with elimination of one sideband except for the 

 wanted vestige. The equivalent performance can be secured by applica- 

 tion of impulses to a band-pass transmission characteristic with the 

 proper interval between pulses in relation to the midband frequency, as 

 discussed below: 



When equation (12.03) is written with respect to the midband fre- 

 quency, Ur = oom , and a sjnnmetrical amplitude characteristic is assumed 

 so that Q = 0, the following relation obtained. 



W{to) = cos (jOmto ^ Ar, cos co„,nTR(to 4- nr) 



(14.10) 



— sin Umfo X^ --In sill comiiTRito + nr), 



n= — 00 



in which R may be replaced by P, the envelope of the impulse charac- 

 teristic. 



Let it be assumed that t is so chosen that cos Umtir = cos n7r/2 in 

 which case sin a)„nr = sin n7r/2. The above equation then becomes 



00 

 W(fo) = cos o^mh X) AnPih + nr) cos mr/2 



— sin (jimh ^ AnP{U) + nr) sin mr/2. 



n=— 00 



The in-phase and quadrature components of the envelope at the sampling 

 instant ^0 = are accordingly 



00 

 i^(0) = Z) AnPinr) cos mr/2, 



(14.12) 



00 



Q(0) = X) AnVim) sin mr/2. 



Pulses in even positions, i.e., A^ , A2 , Ai , etc., will thus contribute an 

 in-phase but no quadrature component while pulses in odd positions 



