THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 



753 



\ho modified ti'ansinissioii-frefiiieiicy cluiracteristic becomes 

 which, inserted in (2.01) gives 



-<'>=^j_: 



ro(/co)e 



ib sin icT iwt 7 



e f/c 



(6.02) 



(6.03) 



are 



(6.04) 



The following relation (Jaeobi's expansion) in which Ji , J2 • 

 Bessel Functions in their usual notation can now be employed 



e = Jo{b) + Ji(h)[e - e J 



J2{b)[e -\- e T 



+ j.me'''" + e-'n + • • • 



Let Po(0 designate the shape of the received pulse or other signal for 

 a transmission frequency characteristic To(i(jo) obtained from (6.03) with 

 6 = 0. In view of (6.04) the solution of (G.03) may then be written 



P{t) = Jo(b)Po(t) + Ji(b)[Po(t + r) - P,{t - r)] 



+ J2{b)[Poit + 2t) + P,(t - 2t)] 



+ J,mPo{t + St) - P,{t - 3r)] 



+ Ji{b)[Po(t + 4r) + Po{t - 4r)] + 



(6.05) 



The shape of the received pulse or signal P{t) is thus obtained by 

 superposing an infinite sequence of pulses or signals of shape Po(t). The 

 peak amplitudes of the pulse or signal echoes and the times at which 

 they occur with respect to ^ = are given in the following table. The 

 reference point t = is arbritrarily selected to coincide with the peak 

 of the pulse Po(t) : 



A sufficient number of echoes must be considered until their peak 

 amplitudes become negligible. 



The superposition of echoes to obtain the resultant pulse is illustrated 

 in Fig. 20. Instead of plotting the various echoes and combining them 



