THEORETICAL FUXDAMEXTALS OF PULSE TKAXS.MlSSlOX 731 



unit step is assumed. For these reasons, only the transmission charac- 

 teristic for impulses will be considered here, or for pulses of sufficiently 

 short duration to be regarded as impulses. 



Corresponding to any transmission-frequency characteristic is an 

 impulse transmission characteristic, P{1), which designates the received 

 pulse as a function of time for a transmitted unit impulse. The impulse 

 and transmission frequency characteristics are interrelated by the follow- 

 ing Fourier integral relations 



P{t) = ~ I TMe'"' do:, (2.01) 



T(ii,) = [ PiOe-'"' dt. (2.02) 



•'—00 



The transmission characteristic for an applied pulse or signal of 

 arbitrary shape G{t) is given by 



H{t) = ~ [ ^(^■a;)>S(^•a;)e'"' c/co, (2.03) 



27r J- 00 



where Siico) is the frequency spectrum of the applied pulse and is given 

 by 



S{io:) = f GiDe""'' dt (2.04) 



J— 00 



In the case of a symmetrical pulse S(io)) is a real function. 

 In view of (1.01), expression (2.03) may also be written 



H{t) = - [ A{oo)S{oo) cos M - '/'(c^)] f/co, (2.05) 



T Jo 



where the relations A (-co) = A(ui), »S( — w) = AS(aj), \l/i — o:) = — '/'(w) 

 have been used, and it is assumed that S(io}) = ».S(co) is a real function, 

 as for a symmetrical pulse. 



In most pulse transmission systems, the applied pulses can be ap- 

 proximated by short rectangular pulses. Rectangular pulses of unit 

 amplitude and duration 8 have a frequency spectrum 



SM^S'-^^. (2.06) 



coo/ Z 



The same pulse transmission characteristic as when an impulse is 

 applied is obtained with a rectangular pulse if A(o)) is modified by the 

 factor (a;5/2)/sin {co8/2). In the following it will be assumed that the 

 applied pulses are of sufficiently short duration to be regarded as im- 



