THEORETICAL Fl'XDAlVIEXTAES OF PULSE TRANSMISSION 737 



mutual interference between the peaks of the received pulses. This is a 

 basic theorem underh'ing the determination of the transmission capacity 

 of idealized systems. 



For an idealized bandpass characteristic between wo and wi , it follows 

 from (2.09) with ^(») = iiTd and '^( — u) = —iiTd that the impulse 

 characteristic with respect to the midband frequency w, = co„, is 



P(t) = 2 cos[co„,^o - M P(t), (3.03) 



where P(t) is given by (3.01) and i/'o = ^m — (^mTd is the phase intercept 

 at zero frequency. For the transmission characteristic to be ideal in the 

 sense that the peak pulse amplitude occurs when ^o = ^ — r^ = 0, it is 

 necessary that \{/o = ±nT, where n is an integer. This is not necessary 

 if the bandwidth is small in relation to the midband frequency. There 

 will then be a large number of cycles of the modulating frequency Um 

 within the envelope P(t), and the latter can be recovered by envelope 

 detection regardless of the phase of the modulating frequency. 

 With \po = zknir, 



P{t) = cos coJo , (3.04) 



X CO Jo 



oJiS sin coi^o ojo8 sin wo^o /„ ^.j-v 



IT Ciilti) IT Uoti 



where Um = (coo + wi)/2 and Wg = (coi — coo)/2. 



The shape of the impulse characteristic as given by (3.04) is illustrated 

 in the upper haK of Fig. 8. Alternately the impulse characteristic may 

 be regarded as made up of two components in accordance with (3.05). 

 The first component corresponds to a low-pass characteristic of band- 

 width coi , the second component to a negative low-pass characteristic 

 of bandwidth ojq , as indicated in the lower part of the Fig. 8. 



The factor sin <j)sto/costo in (3.04) is zero at the same intervals as for a 

 low-pass characteristic of bandwidth cog , as shown in Fig. 8, so that 

 pulses may be transmitted at the same rate without mutual interference 

 between pulse peaks. The bandwidth in the present case, however, is 

 2ajg = oil — coo , so that for the same bandwidth the pulse transmission 

 rate is half as great as for a low-pass characteristic. 



An exception to this is the particular case when wi = 2coo , so that the 

 total bandwidth is wc • The factor sin coo^o/coo^o in (3.05) is then zero at 

 intervals to = l/2/o , while the factor sin ojitn/uifo is zero at intervals 

 l/2/i = l/4/o , as shown in Fig. 9. Pulses may accordingly in principle be 



