990 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



sidebands ^^ith respect to the carrier, equivalent to a band-pass charac- 

 teristic, with the spectrum of the sideband frecjuencies determined by 

 the characteristic of the low-pass filter. The equivalent of an asymmetrical 

 band-pass characteristic can be obtained by suppressing part of the upper 

 or lower sideband with the aid of filters. 



Although the mathematical formulation with both methods is es- 

 sentially the same when Ur is identified with the carrier frequency co<, , 

 with impulse excitation the phase of w^ is fixed in relation to the envelope 

 but is independent of it with carrier modulation. By proper choice of 

 the pulse interval r in (12.03), such that cos Ur (to + nr) = cos UrU or 

 WrT = 27rm, TO = 0, 1, 2 • • • it is possible with impulse excitation to 

 obtain the same relation between the reference or carrier frequency as 

 when the output of a low-pass filter is used to modulate a carrier. In the 

 above case the pulses are transmitted at intervals r = m/fr — m/fc , 

 corresponding to multiples of the duration of a carrier cycle. Since the 

 duration of a carrier cycle is ordinarily small in relation to the pulse 

 interval, there is essentially no important difference in the rate at which 

 pulses can be transmitted with the above two methods. However, with 

 band-pass filters the exact relationship of pulse intervals to the carrier 

 frequency may be difficult to maintain with simple instrumentation, 

 while this is no problem with carrier modulation. For this reason, and 

 since the performance is otherwise equivalent, only the basic relation- 

 ships with, carrier modulation will be discussed further. 



Assuming that cos corik + nr) = cos oorU , as discussed above, equa- 

 tion (12.03) becomes 



W(to) 



00 00 



= cos WrU H A„R{to + nr) + sin WrU XI AnQ{to + nr). (12.07) 



— 00 00 



The envelope at the sampling point is accordingly 



TF(0) 



X! AnRM 



+ 



E AMnr) 



-12X1/2 



(12.08) 



In ideal transmission systems there would be no pulse overlaps or 

 intersymbol interference, and the amplitude of the pulse train at the 

 sampling instant would be 



F(0) = ^o[i^'(0) + Q'(0)]'''. (12.09) 



This condition could be realized with sufficient pulse spacing. However, 

 the objective in the design of efficient pulse systems is to determine the 

 minimum pulse spacing consistent with tolerable intersymbol inter- 



