TIIEOKETICAL FUNDAMENTALS OF PULSE TRANSMISSION 



761 



ponents required to represent a given phase distortion increases, the 

 determination of the resultant pulse becomes rather laborious, unless 

 the sine deviations are all small in amplitude. In the latter case each 

 sine deviation corresponds in a first approximation to a single pair of 

 echoes, so that the effect of a number of sine deviations can be obtained 

 by direct superposition. 



7. PULSE ECHOES FROM AMPLITUDE DISTORTION 



Departures from a given amplitude characteristic may in certain cases 

 be approximated by a single cosine variation, as illustrated in Fig. 26. 

 Since the amplitude characteristic is an even function of co, any departure 

 from a gi\'en amplitude characteristic may be represented by a cosine 

 Fourier series. The effect of a cosine variation in the amplitude charac- 

 teristic is therefore of basic interest. 



A cosine variation will in general be accompanied by a change in the 

 phase characteristic, as discussed in Section 1, but it will first be assumed 

 that phase correction is employed to maintain a fixed phase characteris- 

 tic. 



Let Ao{o)) be the original amplitude characteristic and let the modified 

 amplitude characteristic be of the form 



A(co) = i4o(w)[l + a cos ut] 



(7.01) 



Equation (2.01) for the impulse transmission characteristic then be- 

 fomes, ^nth To(zco) = Ao(w)e~'^"^"\ 



P(0 = ^/ 



ZtT Jco 



To(ico) 



I + -{e 4- e ) 



t iO t J 



e rfoj. 



(7.02) 



= Poit) + I Pod + r) -f I Poit - r). 



(a) RATIO OF AMPLITUDE 

 CHARACTERISTICS 



(b) IMPULSE CHARACTERISTIC 



O 

 I- 

 < 1 



A {cj)/Ao(.co) = 1 +a cos cj T 



FREQUENCY, OJ 



\* T *\* T >] 



Fig. 2G — Pulse echoes from cosine variation in amplitude characteristic with- 

 out change in phase characteristic. 



