TMEORKTICAL FUNDAMENTALS OF PULSE TRANSMISSION 989 



more discrete values as in pulse code modulation systems. In pulse posi- 

 tion modulation, ^4„ = except at the instants pulses of a given ampli- 

 tude are transmitted, and n may not necessarily be an integer. In pulse 

 duration modulation, An = 1 over the intervals iit of varying duration 

 ill which pulses are transmitted, and zero otherwise. Equation (12.02) 

 is thus a general formulation of the wave shape of a received pulse train, 

 applicable to various pulse modulation methods. 



Inserting (2.09) in (12.02) with R^ -\- R+ = R and Q- - Q+ = Q 

 and taking i/'^ = without loss of generality 



W{to) 



00 



= 2Z .-l^fcos 00,(^0 + nT)R{to + ht) + sin cor(/o + nr)Q(/o + nr)], 



= cos O^rto ^ ^n[C0S WrUrRih + Wt) + slll Wr7lTQ{to + Ut)] 



— 00 

 00 



+ sin corto ^ ^„[cos iCrflTQito + Ut) — siii WrUrRih + nr)]. 



(12.03) 



The envelope of the wave at the sampling instant io = is 



Wm = {R' + Q'Y'\ (12.04) 



(12.05) 



R = ^ A„[cos oornrRiriT) + sin ajrnrQ(nr)], 



— 00 

 00 



Q = ^ A„[cos ajrnrQ(nr) — sin ccrnTRinr)]. 



— 00 



For the particular case of a low-pass system 

 Q = 0, and cor = 0, 

 so that 



W{0) = E AnPinr). (12.00) 



n= — 00 



A band-pass characteristic can be obtained with the aid of band-pass 

 filters at the transmitting or receiving ends, or at both ends of a system, 

 and the eciuations developed previously for the impulse characteristic 

 tacitly assumed such an arrangement. Equivalent performance can, 

 however, also be secured by methods which are usually omjiloyed in 

 practice, and to which the equations also apply. Impulses can thus be 

 applied to a low-pass pulse shaping network or filter, and the output used 

 to modulate a carrier. There will then be a svnimctrical distribution of 



