THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 



7G3 



Thus, with II eosino variation in the amjilitudc characteristic in ac- 

 cordance with (7.01), accompanied by a minimum phase shift change in 

 the phase characteristic in accordance with (7.07), the modified impulse 

 characteristic becomes 



Pit) = :r4-, [n(0 + 2rPo(t - r) + r'Po{t - 2r)], (7.10) 



1 -|- r- 



where 



, = i [1 - vr^^^]. 



(7.11) 



The received pulse or signal P(t) will thus consist of three components 

 each having the same shape as the pulse or signal Po{t), but differing in 

 amplitude and displaced in time, as indicated in Fig. 27. 



For small values of the amplitude a of the cosine deviation, r = a/2 

 and I -{- r~ = 1, so that 



Pit) = Poit) + aPoit - r) + ^ Poit - 2r). 



(7.11) 



The solution for a somew^hat similar case given elsewhere,^ has an 

 infinite number of echoes, with the second echo given by a Poit — 2t) 

 rather than (a /4:)Po(t — 2r) as above. In the case referred to, the ampli- 

 tude deviation is in a first approximation a cos cor, but there are addi- 

 tional terms in cos 2oot, cos Scot etc, which are responsible for the different 

 amplitude of the second echo and for the infinite sequence of echoes. 



With a cosine modification in the attenuation characteristic as given 

 by (7.03), there will be a corresponding sine modification in the phase 

 characteristic in accordance with (1.11). The modified transmission- 



(a) RATIO OF AMPLITUDE 

 CHARACTERISTICS 



(b) IMPULSE CHARACTERISTIC 



FREQUENCY, Cv 



Po(t-27-) 



7 *^ r- 



a«i, r = -a/2 



Fig. 27 — Pulse echoes from cosine variation in amplitude characteristic with 

 associated minimum phase shift variation in phase characteristic. 



